i

Modern Antennas and Microwave Circuits

A complete master-level course

by Prof. dr.ir. A.B. Smolders, Prof.dr.ir. H.J. Visser and dr.ir. Ulf Johannsen

Electromagnetics Group (EM) Center for Wireless Technologies Eindhoven (CWT/e) Department of Electrical Engineering Eindhoven University of Technology, The Netherlands

Draft Version 0.5, March 2019 ii Contents

1 Introduction 3 1.1 Purpose of this textbook ...... 3 1.2 Antenna functionality ...... 3 1.3 History of antennas ...... 4 1.4 Electromagnetic spectrum ...... 7

2 System-level antenna parameters 9 2.1 Coordinate system and time-harmonic fields ...... 9 2.2 Field regions ...... 10 2.3 Far field properties ...... 11 2.4 Radiation pattern ...... 13 2.5 Beam width ...... 14 2.6 Directivity and antenna gain ...... 15 2.7 Circuit representation of antennas ...... 15 2.8 Effective antenna aperture ...... 17 2.9 Polarisation properties of antennas ...... 18 2.10 Link budget analysis: basic radio- and radar equation ...... 20

3 Transmission line theory and microwave circuits 23 3.1 Introduction ...... 23 3.2 Telegrapher equation ...... 23 3.3 The lossless transmission line ...... 27 3.4 The terminated lossless transmission line ...... 28 3.5 The quarter-wave transformer ...... 32 3.6 The lossy terminated transmission line ...... 34 3.7 Field analysis of transmission lines ...... 36 3.8 The Smith chart ...... 38 3.9 Microwave networks ...... 40 3.10 Power Combiners ...... 44 3.11 Impedance matching and tuning ...... 51 3.12 Microwave filters ...... 55

iii CONTENTS 1

4 Antenna Theory 59 4.1 Maxwell’s equations ...... 59 4.2 Radiated fields and far-field approximation ...... 65 4.3 Electric dipole ...... 69 4.4 Thin-wire antennas ...... 72 4.5 Wire antenna above a ground plane ...... 80 4.6 Folded dipole ...... 81 4.7 Loop antennas ...... 82 4.7.1 Small loop antennas: magnetic dipole ...... 85 4.7.2 Large loop antennes ...... 86 4.8 Magnetic sources ...... 86 4.9 Lorentz-Lamor theorem ...... 89 4.10 Horn antennas ...... 99 4.11 Reflector antennas ...... 102 4.12 Microstrip antennas ...... 108

5 Numerical electromagnetic analysis 115 5.1 Moment of moments (MoM) ...... 115

6 Phased Arrays and Smart Antennas 123 6.1 Linear phased arrays of isotropic antennas ...... 123 6.1.1 Array factor ...... 123 6.1.2 Uniform amplitude tapering ...... 126 6.1.3 DFT/FFT relation ...... 128 6.1.4 Grating lobes ...... 129 6.1.5 Amplitude tapering and pattern synthesis ...... 129 6.1.6 Directivity and taper efficiency ...... 133 6.1.7 Beam broadening due to beam scanning ...... 135 6.1.8 Phase shifters (PHS) versus true time delay (TTD) ...... 136 6.2 Planar phased arrays of isotropic radiators ...... 138 6.3 Arrays of non-ideal antenna elements ...... 140 6.4 Mutual coupling ...... 142 6.5 The effect of phase and amplitude errors ...... 145 6.6 Noise Figure and Noise Temperature ...... 150 6.7 Sparse arrays ...... 153 6.8 Calibration of phased-arrays ...... 155 6.9 Focal-plane arrays ...... 157 2 CONTENTS Chapter 1

Introduction

1.1 Purpose of this textbook

This textbook provides all relevant material for Master-level courses in the domain of antenna systems. For example, at Eindhoven University of Technology, we use this book for two courses (total of 10 ECTS) in the Electrical Engineering Master program, distributed over a semester. The book includes comprehensive material on antennas and provides introduction-level material on microwave engineering, which is the perfect mixture of know-how for a junior antenna expert. The theoretical material in this textbook can be supplemented by labs in which the students learn how to use state-of-the-art antenna and microwave design tools and test equipment, such as a vector network analyser or a near-field scanner. Several chapters in this book can be used quite independent from each other. For example, chapter 6 can be used in a dedicated phased-array antenna course without the need to use Maxwell-based antenna theory from chapter 4. Similarly, chapter 3 does not require deep back-ground knowledge in electromagnetics or antennas. The antenna theory presented in chapter 4 is based on the original lecture notes of dr. Martin Jeuken [1].

1.2 Antenna functionality

According to the Webster dictionary, an antenna is:

i. one of a pair of slender, movable, segmented sensory organs on the head of insects, myriapods, and crustaceans,

ii. a usually metallic device (such as a rod or wire) for radiating or receiving radio waves

The second definition of an antenna is used within the domain of Electrical Engineering. An antenna transforms an electromagnetic wave in free space in to a guided wave that propagates along a transmission line and vice-versa. In addition, antennas often provide spatial selectivity, which means that the antenna radiates more energy in a certain direction (transmit case) or is more sensitive in a certain direction (receive case). Traditional antenna concepts, like wire antennas, are passive electromagnetic devices which implies that the reciprocity concept holds. As a result, the transmit and receive properties are identical. In integrated antenna concepts,

3 4 CHAPTER 1. INTRODUCTION where the transmit and receive electronics are part of the antenna functionality, reciprocity might not always apply.

1.3 History of antennas

One of the principal characteristics of human beings is that they almost continually send and receive signals to and from one another. The exchange of meaningful signals is the heart of what is called communication. In its simplest form, communication involves two people, namely the signal transmitter and the signal receiver. These signals can take many forms. Words are the most common form. They can be either written or spoken. Before the invention of technical resources such as radio communication or telephone, long-distance communication was very dif- ficult and usually took a lot of time. Proper long-distance communication was at that time only possible by exchange of written words. Couriers were used to transport the message from the sender to the receiver. Since the invention of radio and telephone, far-distance communication or telecommunication is possible, not only of written words, but also of spoken words and almost without any time delay between the transmission and the reception of the signal. Antennas have played an important part in the development of our present telecommunication services. Antennas have made it possible to communicate at far distances, without the need of a physical connection between the sender and receiver. Apart from the sender and receiver, there is a third important element in a communication system, namely the propagation channel. The transmitted signals may deteriorate when they propagate through this channel. In this thesis only the antenna part of a communication system will be investigated. In 1886 Heinrich Hertz, who was a professor of physics at the Technical Institute in Karlsruhe, was the first person who made a complete radio system [3]. When he produced sparks at a gap of the transmitting antenna, sparking also occurred at the gap of the receiving antenna. Hertz in fact visualised the theoretical postulations of James Clerk Maxwell. Hertz’s first experiments used wavelengths of about 8 meters, as illustrated in Fig. 1.1. After Hertz the Italian Guglielmo Marconi became the motor behind the development of practical radio systems [4]. He was not a famous scientist like Hertz, but he was obsessed with the idea of sending messages with a wireless communication system. He was the first who performed wireless communications across the Atlantic. The antennas that Marconi used were very large wire antennas mounted onto two 60-meter wooden poles. These antennas had a very poor efficiency, so a lot of input power had to be used. Sometimes the antenna wires even glowed at night. In later years antennas were also used for other purposes such as radar systems and radio astronomy. In 1953 Deschamps [5] reported for the first time about planar microstrip antennas, also known as patch antennas. It was only in the early eighties that microstrip antennas became an interesting topic among scientists and antenna manufacturers. Microstrip antennas are an example of so-called printed antennas which are metal structures printed on a substrate (e.g. printed-circuit-board (PCB)). Figure 1.2 shows some well- known antenna configurations which are nowadays used in a variety of applications. Reflector antennas, horn antennas, wire antennas and printed antennas all have become mass products. Reflector antennas are often used for the reception of satellite television, whereas wire antennas are commonly used to receive radio signals with a car or portable radio receiver. Printed antennas are also widely used nowadays, for example in smart phones. Printed antennas can be realized 1.3. HISTORY OF ANTENNAS 5

Figure 1.1: First radio system of Heinrich Hertz operating at a wavelength of about 8 meter [3].

Reflector antenna (Westerbork Synthesis Radio Telescope) Wire antenna

Horn antenna Microstrip antenna

Metallic patch

h

εr ground plane

Figure 1.2: Some well-known antenna concepts. 6 CHAPTER 1. INTRODUCTION as two-dimensional structures on PCBs or as three-dimensional structures on plastic substrates. More recently, printed antennas have even been realized on chips in integrated circuits (ICs)[6].

Reflector antennas provide high directivity (spatial selectivity), but have the disadvantage that the main lobe of the antenna has to be steered in the desired direction by means of a highly accurate mechanical steering mechanism. This means that simultaneous communication with several points in space is not possible. Wire and printed antennas are usually more omni-directional, resulting in a low directivity. There are certain applications where these conventional antennas cannot be used. These applications often require a phased-array antenna. A phased-array antenna has the capability to communicate with several targets which may be anywhere in space, simultaneously and continuously, because the main beam of the antenna can be directed electronically into a certain direction. Another advantage of phased-array antennas is the fact that they are relatively flat. Figure 1.3 shows the general phased-array antenna concept. Three essential layers can be

Figure 1.3: Example of a phased-array antenna. The one-square-meter-array (OSMA) was a test- bed of the square-kilometer-array (SKA) and consists of an active center of 64 bow-tie antennas with electric beamforming. The active center is surrounded by two rows of passive bow-tie antennas [7]

distinguished 1) an antenna layer, 2) a layer with transmitter and receiver modules (T/R modules) and 3) a signal-processing and control layer that controls the direction of the main beam of the array. The antenna layer consists of several individual antenna elements which are placed on a rectangular or on a triangular grid. Arrays of antennas can also be used to realized multiple- input-multiple-output (MIMO) communication systems. In such a system several non-correlated spatial communication channels can be realized between two devices (e.g. a base station and mobile user), resulting in a much higher channel capacity [8]. 1.4. ELECTROMAGNETIC SPECTRUM 7

1.4 Electromagnetic spectrum

Antenna systems radiate electromagnetic energy at a particular frequency. The specific frequency range which can be used for a specific application is restricted and regulated worldwide by the International Telecommunication Union (ITU). The ITU organizes World Radiocommunication Conferences (WRCs) to determine the international Radio Regulations, which is the international treaty which defines the use of the world-wide radio-frequency spectrum. An example of such an allocation chart of the radio-frequency spectrum is shown in 1.4. This particular map is valid in the United States, but is very representative for the world-wide spectrum allocation. 4 0 9 1 19.95 20.05 59 61 70 90 110 130 160 190 200 275 285 300

Fixed MARITIME MOBILE MARITIME

FIXED Maritime

Aeronautical (radiobeacons) MARITIME MOBILE MARITIME FIXED MOBILE Radionavigation MOBILE Mobile (radiobeacons)

MARITIME Aeronautical Radionavigation NOT ALLOCATED RADIONAVIGATION FIXED Mobile FIXED MOBILE Aeronautical

MARITIME AERONAUTICAL AERONAUTICAL FIXED RADIONAVIGATION FIXED FIXED RADIONAVIGATION

MARITIME MOBILE MOBILE (radiobeacons) Radiolocation Radiolocation RADIONAVIGATION AERONAUTICAL AERONAUTICAL RADIONAVIGATION STANDARD FREQUENCY AND TIME SIGNAL (60 kHz) TIME SIGNAL AND FREQUENCY STANDARD

UNITED MARITIME RADIONAVIGATION STANDARD FREQUENCY AND TIME SIGNAL (20 kHz) TIME SIGNAL AND FREQUENCY STANDARD

0 kHz 300 kHz

300 325 335 405 415 435 495 505 510 525 535 1605 1615 1705 1800 1900 2000 2065 2107 2170 2173.5 2190.5 2194 2495 2505 2850 3000

MOBILE MOBILE FIXED Aeronautical AERONAUTICAL MOBILE Aeronautical MARITIME MOBILE

Radionavigation AERONAUTICAL RADIONAVIGATION (radiobeacons) RADIONAVIGATION (telephony) (telephony)

MOBILE Radionavigation

(radiobeacons) (radiobeacons) (radiobeacons) MARITIME AERONAUTICAL BROADCASTING MARITIME MOBILE RADIONAVIGATION MARITIME MOBILE except aeronautical mobile (radiobeacons) RADIONAVIGATION AERONAUTICAL RADIONAVIGATION BROADCASTING MOBILE MOBILE MOBILE

Mobile (AM RADIO) except aeronautical mobile except aeronautical mobile AMATEUR MOBILE (R) FIXED Aeronautical BROADCASTING AERONAUTICAL MARITIME MOBILE RADIOLOCATION

STATES MOBILE MARITIME MOBILE MOBILE (distress and calling)

MARITIME MOBILE (distress and calling)

RADIONAVIGATION Aeronautical Maritime

MOBILE RADIO- (radiobeacons) Mobile Mobile MOBILE (ships only) LOCATION (2500kHz) TIME SIGNAL AND FREQ. STANDARD MOBILE Aeronautical Mobile MARITIME MOBILE MARITIME MOBILE Maritime FIXED FIXED AERONAUTICAL FIXED MARITIME MOBILE RADIONAVIGATION (radiobeacons) Radionavigation

Non-Federal Travelers Information Stations (TIS), a mobile service, are authorized in the 535-1705 kHz band. Federal TIS operates at 1610 kHz. 300 kHz 3 MHz

FREQUENCY 3.0 3.025 3.155 3.23 3.4 3.5 4.0 4.063 4.438 4.65 4.7 4.75 4.85 4.995 5.005 5.06 5.45 5.68 5.73 5.59 6.2 6.525 6.685 6.765 7.0 7.1 7.3 7.4 8.1 8.195 8.815 8.965 9.04 9.4 9.9 9.995 10.005 10.1 10.15 11.175 11.275 11.4 11.6 12.1 12.23 13.2 13.26 13.36 13.41 13.57 13.87 14.0 14.25 14.35 14.99 15.01 15.1 15.8 16.36 17.41 17.48 17.9 17.97 18.03 18.068 18.168 18.78 18.9 19.02 19.68 19.8 19.99 20.01 21.0 21.45 21.85 21.924 22.0 22.855 23.0 23.2 23.35 24.89 24.99 25.01 25.07 25.21 25.33 25.55 25.67 26.1 26.175 26.48 26.95 26.96 27.23 27.41 27.54 28.0 29.7 29.8 29.89 29.91

FIXED

FIXED FIXED Mobile Mobile Mobile MOBILE FIXED Mobile FIXED MOBILE FIXED FIXED except aeronautical mobile FIXED FIXED MOBILE FIXED MOBILE except aeronautical mobile MOBILE FIXED LAND MOBILE except aeronautical mobile Mobile except aeronautical mobile except aeronautical mobile (R) except aeronautical mobile (R) except aeronautical mobile (R) AMATEUR SATELLITE AMATEUR MOBILE AMATEUR SATELLITE AMATEUR AMATEUR SATELLITE AMATEUR AMATEUR SATELLITE AMATEUR AMATEUR SATELLITE AMATEUR except aeronautical mobile (R) MOBILE AMATEUR SATELLITE AMATEUR

MARITIME MOBILE MARITIME MOBILE AMATEUR except aeronautical MOBILE MOBILE FIXED FIXED FIXED FIXED FIXED AMATEUR AMATEUR mobile MOBILE MOBILE (R) MOBILE FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED MARITIME AMATEUR MOBILE (R) LAND MOBILE MARITIME LAND MOBILE FIXED LAND MOBILE MOBILE (OR) LAND MOBILE MARITIME MOBILE MARITIME MOBILE MARITIME MOBILE MARITIME MOBILE AERONAUTICAL MARITIME MOBILE BROADCASTING except aeronautical mobile AERONAUTICAL BROADCASTING BROADCASTING FIXED BROADCASTING AERONAUTICAL RADIO ASTRONOMY BROADCASTING BROADCASTING BROADCASTING RADIO ASTRONOMY MOBILE BROADCASTING ALLOCATIONS BROADCASTING MARITIME MOBILE BROADCASTING AERONAUTICAL MOBILE (R) AERONAUTICAL AERONAUTICAL MOBILE (R) AERONAUTICAL

MOBILE MOBILE (OR) AERONAUTICAL

MARITIME MOBILE Mobile

except aeronautical MOBILE (OR) AERONAUTICAL AERONAUTICAL MOBILE (OR) AERONAUTICAL AERONAUTICAL MOBILE (OR) AERONAUTICAL AERONAUTICAL MOBILE (OR) AERONAUTICAL AERONAUTICAL MOBILE (OR) AERONAUTICAL except aeronautical mobile (R) except MOBILE (OR) AERONAUTICAL MOBILE MOBILE AMATEUR Mobile AMATEUR AMATEUR mobile aeronautical mobile MOBILE (R) AERONAUTICAL MOBILE AERONAUTICAL MOBILE (R) AERONAUTICAL AERONAUTICAL MOBILE (R) AERONAUTICAL FIXED MOBILE AERONAUTICAL MOBILE (R) AERONAUTICAL AERONAUTICAL MOBILE (R) AERONAUTICAL AMATEUR MOBILE AMATEUR AERONAUTICAL MOBILE (R) AERONAUTICAL FIXED AERONAUTICAL MOBILE (R) AERONAUTICAL AMATEUR AERONAUTICAL MOBILE (OR) AERONAUTICAL FIXED FIXED FIXED FIXED AERONAUTICAL MOBILE (OR) AERONAUTICAL FIXED FIXED FIXED FIXED FIXED Radiolocation (R) FIXED STANDARD FREQ. AND TIME SIGNAL (25 MHz) TIME SIGNAL AND FREQ. STANDARD MARITIME MOBILE FIXED FIXED except aeronautical mobile (R) FIXED STANDARD FREQUENCY AND TIME SIGNAL (5 MHz) TIME SIGNAL AND FREQUENCY STANDARD STANDARD FREQUENCY AND TIME SIGNAL (20 MHz) TIME SIGNAL AND FREQUENCY STANDARD STANDARD FREQUENCY AND TIME SIGNAL (10 MHz) TIME SIGNAL AND FREQUENCY STANDARD except aeronautical mobile (R) except aeronautical mobile (R) except aeronautical mobile (R) except aeronautical mobile (R) STANDARD FREQUENCY AND TIME SIGNAL (15 MHz) TIME SIGNAL AND FREQUENCY STANDARD

THE RADIO SPECTRUM 3MHz ISM - 6.78 ± .015 MHz ISM - 13.560 ± .007 MHz ISM - 27.12 ± .163 MHz 30 MHz 30.0 30.56 32.0 33.0 34.0 35.0 36.0 37.0 37.5 38.0 38.25 39.0 40.0 42.0 43.69 46.6 47.0 49.6 50.0 54.0 72.0 73.0 74.6 74.8 75.2 75.4 76.0 88.0 108.0 117.975 121.9375 123.0875 123.5875 128.8125 132.0125 136.0 137.0 137.025 137.175 137.825 138.0 144.0 146.0 148.0 149.9 150.05 150.8 152.855 154.0 156.2475 156.7625 156.8375 157.0375 157.1875 157.45 161.575 161.625 161.775 161.9625 161.9875 162.0125 163.0375 173.2 173.4 174.0 216.0 217.0 219.0 220.0 222.0 225.0 300.0 Fixed FIXED Amateur (space-to-Earth) (space-to-Earth) (space-to-Earth) (space-to-Earth) MET. SATELLITE MET. MET. SATELLITE MET. SATELLITE MET. MET. SATELLITE MET. FIXED RADIO SERVICES COLOR LEGEND MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE MOBILE LAND MOBILE LAND MOBILE LAND MOBILE (Earth-to-space) Land mobile AMATEUR- SATELLITE AMATEUR- LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE MOBILE-SATELLITE LAND MOBILE AERONAUTICAL Land mobile MOBILE except (space-to-Earth) (space-to-Earth) (space-to-Earth) (space-to-Earth) aeronautical mobile

INTER-SATELLITE RADIO ASTRONOMY OPERATION SPACE SPACE OPERATION SPACE MOBILE OPERATION SPACE OPERATION SPACE BROADCASTING MOBILE except

MOBILE BROADCASTING AERONAUTICAL FIXED AMATEUR BROADCASTING BROADCASTING aeronautical mobile

(TELEVISION) AMATEUR AMATEUR (TELEVISION ) (TELEVISION) (FM RADIO) RADIONAVIGATION FIXED LAND MOBILE FIXED MOBILE (R) MARITIME MOBILE MARITIME MOBILE MOBILE (R) MOBILE (R) MARITIME MOBILE MARITIME MOBILE LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE AERONAUTICAL MOBILE AERONAUTICAL MARITIME MOBILE (AIS) MARITIME MOBILE (AIS) (space-to-Earth) (space-to-Earth) (space-to-Earth) (space-to-Earth) AERONAUTICAL MOBILE AERONAUTICAL AERONAUTICAL FIXED AERONAUTICAL AERONAUTICAL RADIO ASTRONOMY SPACE RESEARCH SPACE SPACE RESEARCH SPACE RESEARCH SPACE SPACE RESEARCH SPACE AERONAUTICAL MOBILE (R) AERONAUTICAL AERONAUTICAL RADIODETERMINATION MOBILE (R) AERONAUTICAL FIXED Mobile FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED

LAND MOBILE FIXED AMATEUR

FIXED FIXED MOBILE except aeronautical mobile FIXED MOBILE except aeronautical mobile FIXED FIXED MOBILE except aeronautical mobile FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED SATELLITE FIXED MOBILE SATELLITE SATELLITE mobile AERONAUTICAL RADIONAVIGATION AERONAUTICAL MOBILE (Earth-to-space) RADIONAVIGATION- MOBILE-SATELLITE Mobile-satellite Radio astronomy Mobile-satellite (space-to-Earth) (space-to-Earth) (space-to-Earth) (space-to-Earth) except aeronautical Land mobile Fixed MOBILE-SATELLITE MOBILE-SATELLITE MARITIME MOBILE (distress, urgency, safety and calling) MARITIME MOBILE (distress, urgency, RADIO ASTRONOMY LAND MOBILE AERONAUTICAL RADIOLOCATION SATELLITE RADIONAVIGATION 30 MHz ISM - 40.68 ± .02 MHz 300 MHz 300.0 328.6 335.4 399.9 400.05 400.15 401.0 402.0 403.0 406.0 406.1 410.0 420.0 450.0 454.0 455.0 456.0 460.0 462.5375 462.7375 467.5375 467.7375 470.0 512.0 608.0 614.0 698.0 763.0 775.0 793.0 805.0 806.0 809.0 849.0 851.0 854.0 894.0 896.0 901.0 902.0 928.0 929.0 930.0 931.0 932.0 935.0 940.0 941.0 944.0 960.0 1164.0 1215.0 1240.0 1300.0 1350.0 1390.0 1392.0 1395.0 1400.0 1427.0 1429.5 1430.0 1432.0 1435.0 1525.0 1559.0 1610.0 1610.6 1613.8 1626.5 1660.0 1660.5 1668.4 1670.0 1675.0 1695.0 1710.0 1761.0 1780.0 1850.0 2000.0 2020.0 2025.0 2110.0 2180.0 2200.0 2290.0 2300.0 2305.0 2310.0 2320.0 2345.0 2360.0 2390.0 2395.0 2417.0 2450.0 2483.5 2495.0 2500.0 2655.0 2690.0

AMATEUR RADIOLOCATION SATELLITE Space research Amateur MARITIME MOBILE (active) FIXED FIXED (line of sight only) Fixed Fixed (S-E) (E-S) (E-S) MET. SAT. MET. Fixed-satellite FIXED (space-to-Earth) Earth Expl Sat Earth FIXED FIXED FIXED RADIOLOCATION FIXED FIXED FIXED

Space research (Earth-to-space) (Earth-to-space)

Earth Expl Sat BROADCASTING FIXED exploration- (passive) EARTH Fixed (active) MOBILE SATELLITE MOBILE SATELLITE (TELEVISION) satellite MOBILE Mobile Mobile

MOBILE ** MOBILE SPACE RESEARCH SPACE (space-to-Earth) EXPLORATION- MOBILE FIXED (telemetry) (Earth-to-space)

Radiolocation (ling of sight only

(TELEVISION) (active) MOBILE SATELLITE

MOBILE SATELLITE SATELLITE AERONAUTICAL

MOBILE including aeronautical RADIOLOC ATION (E-S) RADIOLOCATION Fixed-satellite (Earth-to-space) FIXED RADIONAVIGATION BROADCASTING MOBILE **

(space-to-space) Earth

MOBILE ** telemetry, but excluding (radiosonde)

MOBILE (passive) FIXED FIXED MOBILE Met-Satellite SPACE RESEARCH SPACE

Radio- Radio- MOBILE BROADCASTING BROADCASTING location location MOBILE MOBILE exploration- flight testing of manned MOBILE Amateur MOBILE ** MOBILE MOBILE (Earth-to-space) MOBILE** AMATEUR Radiolocation (E-S)

SPACE MOBILE AERONAUTICAL (space-to-Earth) (Earth-to-space) satellite aircraft) LAND MOBILE LAND MOBILE AERONAUTICAL MOBILE SATELLITE (S-E) AERONAUTICAL RADIONAVIGATION Met-Satellite MOBILE SATELLITE LAND MOBILE LAND MOBILE MARITIME MOBILE RESEARCH AIDS METEOROLOGICAL RADIONAVIGATION Fixed MOBILE SATELLITE (active) LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE MOBILE** LAND MOBILE LAND MOBILE Space Opn. RADIONAVIGATION AERONAUTICAL AERONAUTICAL

(active) RADIONAVIGATION RADIONAVIGATION MOBILE**

AMATEUR SATELLITE BROADCASTING - SATELLITE

RADIO ASTRONOMY AERONAUTICAL LAND MOBILE RADIO ASTRONOMY SPACE LAND MOBILE LAND MOBILE LAND MOBILE SPACE RESEARCH (passive) SPACE

SATELLITE BROADCASTING Fixed EXPL EXPL EARTH EARTH

BROADCASTING OPERATION MOBILE SAT. (E-S) SAT. RADIONAVIGATION (telemetry & telecommand) SPACE FIXED LAND MOBILE (telemetry & telecommand) (Earth-to-space) FIXED FIXED FIXED (space-to-space) (space-to-Earth) (TELEVISION) (Earth-to-space) MOBILE SATELLITE (TELEVISION) FIXED Radiolocation SATELLITE SATELLITE FIXED RESEARCH

FIXED OPERATION SPACE (S-E)

(space-to-space) BROADCASTING BROADCASTING RADIO MOBILE (active) AIDS MOBILE AERONAUTICAL Amateur SPACE RES. SPACE MOBILE** MOBILE AMATEUR RADIONAVIGATION MOBILE ASTRONOMY MOBILE SAT. (E-S) SAT. (E-S) (400.1 MHz) RADIO ASTRONOMY SATELLITE Radio EARTH EARTH EXPL MET-SAT. (space-to-Earth) (space-to-Earth) RADIOLOCATION Fixed LAND MOBILE (space-to-space) EARTH EARTH LAND MOBILE FIXED LAND MOBILE LAND MOBILE FIXED LAND MOBILE telemetry and telecommand) RADIONAVIGATION- MOBILE SATELLITE METEOROLOGICAL METEOROLOGICAL ( astronomy RADIO- EXPLORATION- EXPLORATION- AERONAUTICAL MOBILE AERONAUTICAL AERONAUTICAL MOBILE AERONAUTICAL Meteorological LOCATION MOBILE ** SATELLITE SATELLITE RADIO- (Earth-to-space) (space-to-Earth) RADIOLOCATION RADIODETERMINATION- RADIODETERMINATION- mand) mand) FIXED

MARITIME (E-S) SATELLITE (Earth-to-space) SATELLITE (Earth-to-space) SATELLITE MOBILE MOBILE

MOBILE Space research FIXED BROADCASTING (S-E) SAT FIXED RADIONAVIGATION SATELLITE MET-SAT. Satellite LOCATION (space-to-space) (space-to-space) FIXED

FIXED FIXED EARTH

(S-E) MARITIME FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED MOBILE (aeronautical telemetry) FIXED (passive) MOBILE**

EXPLORATION- MOBILE SATELLITE(Earth-to-space) MOBILE SATELLITE (space-to-Earth) MOBILE SATELLITE FIXED FIXED (radiosonde) (space-to-Earth) MOBILE

RADIONAVIGATION (Earth-to-space) OPERATION SPACE (passive) MOBILE MOBILE MOBILE SPACE OPN. SPACE SATELLITE Amateur

telecom telecom RADIO- medical telecommand) FIXED AERONAUTICAL RADIONAVIGATION AERONAUTICAL LAND MOBILE LAND MOBILE (medical telemetry and FIXED MOBILE SATELLITE (Earth-to-space) MOBILE SATELLITE Radiolocation

FIXED SPACE SPACE METEOROLOGICAL METEOROLOGICAL METEOROLOGICAL (passive)

EARTH (active) FIXED MOBILE FIXED METEOROLOGICAL AIDS METEOROLOGICAL (RADIOSONDE)

METEOROLOGICAL AIDS METEOROLOGICAL RESEARCH RESEARCH Earth exploration- FIXED (telemetry and FIXED (telemetry and

LAND MOBILE (medical telemetry and medical telecommand) FIXED

EXPLORATION- RADIODETERMINATION- RADIO NAVIGATION (deep space) SATELLITE (space-to-Earth) SATELLITE (space-to-Earth) SATELLITE SATELLITE (space-to-Earth) SATELLITE RADIODETERMINATION- EARTH EXPLORATION - SATELLITE EXPLORATION EARTH RADIO (space-to-Earth) (Earth-to-space) (space-to-Earth) RESEARCH SPACE SATELLITE RADIONAVIGATION AERONAUTICAL satellite AERONAUTICAL (Earth-to-space) SATELLITE SPACE RESEARCH SPACE Amateur RADIO ASTRONOMY RADIO ASTRONOMY Radiolocation RADIOLOCATION RADIONAVIGATION-SATELLITE (space-to-Earth)(space-to-space) MET. AIDS MET. RADIO - Mobile-satellite RADIONAVIGATION-SATELLITE (space-to-Earth)(space-to-space) RADIOLOCATION RADIOLOCATION (active) RADIODETERMINATION- MET. AIDS MET. medical telecommand Radiolocation (Radiosonde) MET. AIDS MET. (space-to-Earth) LAND MOBILE (medical telemetry and (space-to-space) (space-to-space) ASTRONOMY (passive) ASTRONOMY FIXED RADIONAVIGATION SATELLITE RADIONAVIGATION STANDARD FREQUECY AND TIME SIGNAL - SATELLITE - SATELLITE TIME SIGNAL AND FREQUECY STANDARD MOBILE SATELLITE (space-to-Earth) SATELLITE (Radiosonde) SPACE OPERATION (Earth-to-space) OPERATION SPACE (Radiosonde) NAVIGATION BROADCASTING METEOROLOGICAL SPACE OPERATION ISM - 915.0± .13 MHz SATELLITE 300 MHz ISM - 2450.0± .50 MHz 3 GHz

EARTH EXPLORATION METEOROLOGICAL SPACE RESEARCH SATELLITE SATELLITE 3.0 3.1 3.3 3.5 3.6 3.65 3.7 4.2 4.4 4.5 4.8 4.94 4.99 5.0 5.01 5.03 5.15 5.25 5.255 5.35 5.46 5.47 5.57 5.6 5.65 5.83 5.85 5.925 6.425 6.525 6.7 6.875 7.025 7.075 7.125 7.145 7.19 7.235 7.25 7.3 7.45 7.55 7.75 7.85 7.9 8.025 8.175 8.215 8.4 8.45 8.5 8.55 8.65 9.0 9.2 9.3 9.5 9.8 10.0 10.45 10.5 10.55 10.6 10.68 10.7 11.7 12.2 12.7 13.25 13.4 13.75 14.0 14.2 14.4 14.5 14.7145 14.8 15.1365 15.35 15.4 15.43 15.63 15.7 16.6 17.1 17.2 17.3 17.7 17.8 18.3 18.6 18.8 19.3 19.7 20.2 21.2 21.4 22.0 22.21 22.5 22.55 23.55 23.6 24.0 24.05 24.25 24.45 24.65 24.75 25.05 25.25 25.5 27.0 27.5 Earth Earth Earth Earth Earth Earth Earth Earth EARTH RADIO - EARTH EARTH EARTH exploration- exploration- exploration- exploration- exploration- exploration - exploration - exploration - EXPLORATION LOCATION EXPLORATION- EXPLORATION- EXPLORATION - Fixed Fixed

satellite satellite satellite SATELLITE SATELLITE FIXED

FIXED SATELLITE FIXED FIXED FIXED FIXED FIXED FIXED satellite (active) satellite (active) satellite (active) satellite (active) satellite (active) FIXED FIXED SATELLITE Space FIXED satellite -to-Earth) (active) (active) (active) (active) (active) (passive) Radio- location (no airborne) MOBILE Mobile-satellite

(Earth-to-space) (space-to-Earth)

FIXED research FIXED Space research Radiolocation Radiolocation Radiolocation Space research Amateur

FIXED (Earth-to-space)

Space RADIO - Radiolocation Mobile Radio- FIXED-SATELLITE RADIO- Radio- FIXED Radiolocation Radiolocation Radiolocation Radiolocation Amateur

Radiolocation FIXED-SATELLITE Space research (active) location LOCATION MOBILE Amateur Space research location LOCATION MOBILE** Amateur STANDARD FREQUENCY Space research Amateur- research SPACE (Earth-to-space) Space research Space research FIXED-SATELLITE MOBILE FIXED FIXED Radiolocation SPACE FIXED- MOBILE FIXED FIXED MOBILE** FIXED FIXED MOBILE (space-to-Earth) (active) (active) INTER-SATELLITE RADIOLOCATION (no airborne) (no airborne) (active)

(active) MOBILE Radiolocation RESEARCH SATELLITE (passive) SATELLITE Radiolocation Radiolocation Radiolocation MOBILE MOBILE Mobile-satellite Mobile-satellite Amateur MOBILE MOBILE (Earth-to-space) (Earth-to-space) MOBILE** RADIOLOCATION (Earth-to-space)

AND TIME SIGNAL Radiolocation Mobile-satellite

Space INTER-SATELLITE EARTH EXPLORATION - EXPLORATION EARTH (Earth-to-space) SATELLITE SPACE

FIXED Space (space-to-Earth) (Earth-to-space)

RESEARCH FIXED-SATELLITE (Earth-to-space) (Earth-to-space) EARTH (Earth-to-space) (space-to-Earth) FIXED-SATELLITE FIXED SATELLITE FIXED SATELLITE FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE FIXED-SATELLITE FIXED-SATELLITE research Earth-to-space) RESEARCH (passive) SATELLITE research EARTH ( (Earth-to-space) EXPLORATION- - (passive) SATELLITE Space research (deep space)(space-to-Earth) Amateur EARTH EXPLORATION - EXPLORATION EARTH EARTH EXPLORATION - EXPLORATION EARTH EARTH EXPLORATION-SATELLITE (passive) EXPLORATION-SATELLITE EARTH INTER-SATELLITE

Earth RADIOLOCATION RADIONAVIGATION-SATELLITE EARTH EARTH EARTH EARTH FIXED (active) EXPLORATION - (active) MOBILE-SATELLITE Amateur

SATELLITE (active) AMATEUR-SATELLITE Mobile-satellite FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE Space research (deep space)(Earth-to-space) FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE (Earth-to-space) Space Research (passive) (Earth-to-space) RADIONAVIGATION-SATELLITE RADIONAVIGATION-SATELLITE (space-to-Earth)(space-to-space) EXPLORATION- EXPLORATION- EXPLORATION- EXPLORATION- (space-to-Earth) exploration -

FIXED-SATELLITE Space INTER-SATELLITE SATELLITE (Earth-to-space) FIXED-SATELLITE (active) MARITIME RADIONAVIGATION (space-to-Earth) FIXED-SATELLITE INTER-SATELLITE MOBILE-SATELLITE (space MOBILE-SATELLITE FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE SATELLITE SATELLITE SATELLITE SATELLITE (Earth-to-space) FIXED-SATELLITE EARTH (active) satellite research SPACE RESEARCH MOBILE Earth MOBILE (active) (active) (active) (active) EARTH (Earth-to-space) RADIOLOCATION-SATELLITE

MOBILE-SATELLITE (space-to-Earth) MOBILE-SATELLITE (active) Mobile-satellite (space-to-Earth) AIDS Mobile-satellite (space-to-Earth) FIXED FIXED EXPLORATION MOBILE FIXED exploration- FIXED MOBILE-SATELLITE (Earth-to-space) MOBILE-SATELLITE SPACE (Earth-to-space) MOBILE (Earth-to-space) FIXED Meteorological Aids

RADIOLOCATION (active) RADIOLOCATION METEOROLOGICAL Fixed FIXED SATELLITE (space-to-Earth) FIXED-SATELLITE EXPLORATION- (passive) Mobile SPACE RESEARCH SPACE RESEARCH SPACE RESEARCH Amateur-satellite

SPACE RESEARCH MOBILE EARTH EXPLORATION- EXPLORATION- EARTH MOBILE-SATELLITE (space-to-Earth) MOBILE-SATELLITE SATELLITE SATELLITE SATELLITE satellite (active) MOBILE SATELLITE (passive) SATELLITE (space-to-Earth) Radio- SATELLITE (space-to-Earth) SATELLITE

EARTH EXPLORATION-SATELLITE (passive) EXPLORATION-SATELLITE EARTH RESEARCH FIXED-SATELLITE

STANDARD FREQUENCY AND MOBILE RADIOLOCATION SATELLITE (active) (active) (active) - EXPLORATION EARTH (active) (active) Radiolocation METEOROLOGICAL METEOROLOGICAL location RADIOLOCATION

MOBILE SATELLITE Radiolocation FIXED SATELLITE (active) Radio - RESEARCH SPACE SPACE RESEARCH (Earth-to-space) SPACE AERONAUTICAL (active) Radiolocation (passive) FIXED SPACE RESEARCH SPACE RESEARCH SPACE Mobile Earth SPACE (space-to-Earth) TIME SIGNAL SATELLITE location Radio- (passive) FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE

Space RADIO- FIXED-SATELLITE

RADIONAVIGATION RADIOLOCATION RADIOLOCATION FIXED exploration- Standard RESEARCH FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE FIXED

RADIO- LOCATION SPACE RESEARCH (deep space)(Earth-to-space) SPACE RADIOLOCATION RADIOLOCATION SPACE RESEARCH (passive) SPACE RADIO- research location RESEARCH SPACE

satellite (Earth-to-space)(space-to-Earth) FIXED-SATELLITE

FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE frequency

(Earth-to-space) LOCATION (active) FIXED-SATELLITE Radiolocation FIXED

FIXED LOCATION Mobile-satellite (Earth-to-space) FIXED FIXED FIXED and FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE Inter-satellite AERONAUTICAL RADIONAVIGATION AERONAUTICAL MOBILE AERONAUTICAL RADIONAVIGATION AERONAUTICAL time signal AMATEUR FIXED-SATELLITE AERONAUTICAL (Earth-to-space) Standard frequency SPACE RESEARCH (space-to-Earth) SPACE (passive) RADIOLOCATION

RADIOLOCATION Earth

(Earth-to-space) Space RADIOLOCATION FIXED-SATELLITE FIXED-SATELLITE FIXED FIXED-SATELLITE FIXED-SATELLITE MARITIME

RADIONAVIGATION AERONAUTICAL Fixed SPACE RESEARCH (deep space)(space-to-Earth) SPACE (space-to-Earth) FIXED (Earth-to-space) METEOROLOGICAL- SPACE SPACE AERONAUTICAL RADIONAVIGATION AERONAUTICAL satellite

and time signal Radiolocation

(space-to-Earth) RADIONAVIGATION MARITIME exploration - MARITIME RADIONAVIGATION FIXED Standard frequency and Fixed RADIO (active) FIXED-SATELLITE

research FIXED FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE (passive) INTER-SATELLITE RADIONAVIGATION MOBILE Aeronatuical AERONAUTICAL SPACE RESEARCH SPACE FIXED RADIONAVIGATION FIXED BROADCASTING-SATELLITE RADIO EARTH EXPLORATION- EARTH BROADCASTING-SATELLITE

SATELLITE (space-to-Earth) SATELLITE RESEARCH RADIO ASTRONOMY RESEARCH (space-to- RADIONAVIGATION Space research (ground based) (ground based) satellite satellite Radionavigation RADIONAVIGATION FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE SATELLITE

MOBILE time signal satellite FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE AERONAUTICAL MOBILE RADIONAVIGATION RADIONAVIGATION ASTRONOMY AERONAUTICAL RADIOLOCATION SATELLITE (space-to-Earth) SATELLITE

Mobile-satellite (space-to-Earth) (active) FIXED-SATELLITE (Earth-to-space)(space-to-Earth) FIXED-SATELLITE FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE RADIOLOCATION RADIOLOCATION RADIONAVIGATION MOBILE** SPACE RESEARCH (passive) SPACE Space RADIONAVIGATION AERONAUTICAL RADIONAVIGATION AERONAUTICAL (space-to-Earth) MARITIME RADIONAVIGATION AERONAUTICAL RADIONAVIGATION AERONAUTICAL AERONAUTICAL RADIONAVIGATION AERONAUTICAL FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE RADIOLOCATION research Radiolocation RADIOLOCATION RADIOLOCATION Radiolocation ASTRONOMY FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE Earth) RADIONAVIGATION SPACE RESEARCH SPACE

(active) RADIO ASTRONOMY (active) EARTH EXPLORATION-SATELLITE EARTH (active) FIXED-SATELLITE (space-to-Earth) FIXED-SATELLITE FIXED RADIOLOCATION

(Earth-to-space) RADIO ASTRONOMY RADIOLOCATION FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE METEOROLOGICAL- Inter-satellite ACTIVITY CODE RADIOLOCATION (Earth-to-space)

FEDERAL EXCLUSIVE FEDERAL/NON-FEDERAL SHARED 3 GHz 30 GHz 30.0 31.0 31.3 31.8 32.3 33.0 33.4 34.2 34.7 35.5 36.0 37.0 37.5 38.0 38.6 39.5 40.0 40.5 41.0 42.0 42.5 43.5 45.5 46.9 47.0 47.2 48.2 50.2 50.4 51.4 52.6 54.25 55.78 56.9 57.0 58.2 59.0 59.3 64.0 65.0 66.0 71.0 74.0 76.0 77.0 77.5 78.0 81.0 84.0 86.0 92.0 94.0 94.1 95.0 100.0 102.0 105.0 109.5 111.8 114.25 116.0 122.25 123.0 130.0 134.0 136.0 141.0 148.5 151.5 155.5 158.5 164.0 167.0 174.5 174.8 182.0 185.0 190.0 191.8 200.0 209.0 217.0 226.0 231.5 232.0 235.0 238.0 240.0 241.0 248.0 250.0 252.0 265.0

FIXED NON-FEDERAL EXCLUSIVE Fixed FIXED FIXED ISM – 5.8 ± .075 GHz FIXED- FIXED- EARTH FIXED- FIXED FIXED ISM –FIXED 24.125 ± 0.125 SATELLITE SATELLITE EXPLORATION- FIXED SATELLITE EARTH MOBILE Amateur Mobile MOBILE FIXED EARTH SPACE RADIO- EARTH FIXED FIXED 3GHz FIXED FIXED (Earth-to-space) (passive) FIXED MOBILE FIXED (Earth-to-space) EXPLORATION-

FIXED (space-to-Earth) INTER- INTER- INTER- SATELLITE INTER- INTER- LOCATION

MOBILE Amateur SATELLITE

MOBILE RESEARCH INTER- (Earth-to-space) EXPLORATION - EXPLORATION MOBILE MOBILE EARTH RADIO

EXPLORATION- RADIO

RADIO MOBILE INTER- SATELLITE SATELLITE SATELLITE Amateur SATELLITE SATELLITE SATTELLITE (active) SATTELLITE AMATEUR Amateur Amateur (passive) RADIO SATELLITE EARTH FIXED- EARTH EARTH SATELLITE EARTH (Earth-to-space) FIXED SATELLITE (passive) (passive)

EARTH MOBILE- EARTH EXPLORATION EARTH EARTH

(space-to-Earth) (active) (passive) EARTH FIXED-SATELLITE (passive) RADIO (passive) (passive) SATELLITE

MOBILE- SATELLITE FIXED-SATELLITE SATELLITE ASTRONOMY (passive) EXPLORATION- EXPLORATION- SATELLITE

MOBILE SATELLITE SATELLITE ASTRONOMY AMATEUR ASTRONOMY INTER- SATELLITE SATELLITE MOBILE - (passive) MOBILE ASTRONOMY (Passive) (space-to-Earth) EXPLORATION- EXPLORATION-

SATELLITE EXPLORATION- EXPLORATION- SATELLITE EXPLORATION - EXPLORATION SATELLITE RADIO (passive) SATELLITE EXPLORATION- FIXED-

SATELLITE EXPLORATION- EARTH EXPLORATION - EXPLORATION EARTH RADIOLOCATION SATELLITE RADIOLOCATION BROADCASTING SPACE RESEARCH SPACE SPACE RESEARCH SPACE EXPLORATION- ASTRONOMY Amateur - satellite

(passive) (passive) SATELLITE SATELLITE (passive) SATELLITE SATTELLITE (passive) SATTELLITE SPACE RESEARCH SPACE SATELLITE (passive) SATELLITE

LOCATION BROADCASTING (Earth-to-space) Radiolocation

(space-to-Earth) SATELLITE (Earth-to-space) ALLOCATION USAGE DESIGNATION (passive) SATELLITE EARTH EARTH EXPLORATION- FIXED

MOBILE (Earth-to-space) FIXED EARTH EXPLORATION- EARTH MOBILE-SATELLITE Amateur-satellite EARTH EXPLORATION- EARTH satellite (space-to-Earth) SPACE MOBILE** Amateur- SPACE RESEARCH (passive) SPACE SPACE RESEARCH (passive) SPACE (space-to-Earth) Amateur-satellite SPACE MOBILE SPACE RESEARCH SPACE Radiolocation Radiolocation MOBILE SATELLITE SATELLITE Radiolocation MOBILE

MOBILE RESEARCH satellite MOBILE

RESEARCH MOBILE (passive) SPACE RESEARCH (passive) SPACE SATELLITE (passive) SATELLITE AMATEUR - SATELLITE AMATEUR EARTH EXPLORATION-SATELLITE (passive) EXPLORATION-SATELLITE EARTH (passive) EXPLORATION-SATELLITE EARTH RADIO AMATEUR-SATELLITE THIS CHART WAS CREATED (passive) EXPLORATION-SATELLITE EARTH FIXED SATELLITE SATELLITE (passive) SATELLITE RADIOLOCATION RADIO RADIO (space-to-Earth) RADIO SATELLITE EARTH EXPLORATION- EXPLORATION- EARTH

RADIO- EARTH BROADCASTING-

BROADCASTING- RADIO (active) RADIO- (space-to-Earth) RADIONAVIGATION Earth exploration (active) EXPLORATION- EARTH

INTER-SATELLITE (active) NAVIGATION FIXED-SATELLITE FIXED- SPACE SATELLITE FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE FIXED- RADIO- SPACE RESEARCH SPACE RADIO- SERVICE EXAMPLE DESCRIPTION ASTRONOMY SPACE ASTRONOMY NAVIGATION EARTH EXPLORATION- EARTH FIXED Amateur-satellite

(passive) LOCATION

ASTRONOMY ASTRONOMY EXPLORATION- Amateur BY SATELLITE SATELLITE LOCATION SPACE RESEARCH FIXED- LOCATION RESEARCH RADIO

SPACE (passive) (space-to-Earth) SPACE (passive) RADIO FIXED- MOBILE- (space-to-Earth) RADIO (deep space) (Earth-to-space) SATELLITE SPACE RESEARCH SPACE (space-to- RESEARCH RADIO RADIO SATELLITE EARTH ASTRONOMY SATELLITE RADIO (space-to-Earth) (Earth-to-space) MOBILE ASTRONOMY (passive) RADIO ASTRONOMY FIXED - RESEARCH (Passive) ASTRONOMY FIXED RESEARCH (passive) (space-to-Earth)

Earth) SPACE MOBILE RADIO SPACE SPACE RESEARCH SPACE FIXED (Earth-to-space) SATELLITE

DELMON(passive) C. MORRISON (passive)

MOBILE EARTH SPACE AMATEUR-SATELLITE

INTER- RESEARCH SPACE ASTRONOMY ASTRONOMY Primary FIXED Capital Letters FIXED-SATELLITE RADIO- (passive) SPACE RESEARCH (passive) SPACE INTER- EXPLORATION- EXPLORATION- RESEARCH (Earth-to-space) Space ASTRONOMY RADIO- RESEARCH RADIO (Earth-to-space) (passive) SPACE RESEARCH SPACE Earth RESEARCH RADIO RADIOLOCATION

BROADCASTING SATELLITE ASTRONOMY MOBILE

SATELLITE (passive) SATELLITE INTER-SATELLITE EARTH EXPLORATION-SATELLITE (passive) EXPLORATION-SATELLITE EARTH SATELLITE RADIO ASTRONOMY LOCATION SATELLITE SPACE RESEARCH SPACE RADIO ASTRONOMY INTER- INTER- EXPLORATION- EXPLORATION-

RADIO- AMATEUR RADIO INTER- NAVIGATION- research (space-to-Earth) NAVIGATION- SPACE RESEARCH (passive) SPACE EARTH EXPLORATION-SATELLITE EXPLORATION-SATELLITE EARTH NOT ALLOCATED NOT exploration -

JUNE 1, 2011 (Earth-to-space) MOBILE-SATELLITE Space (space-to-Earth) SATELLITE (passive) SATELLITE ASTRONOMY (Earth-to-space) sattellite (active) FIXED-SATELLITE ASTRONOMY

(space-to- MOBILE INTER- SATELLITE RADIO SATELLITE SATELLITE LOCATION SATELLITE

EARTH EXPLORATION-SATELLITE (passive) EXPLORATION-SATELLITE EARTH (space-to-Earth) MOBILE RADIO- Secondary Mobile 1st Capital with lower case letters MOBIL-ESATELLITE research SATELLITE MOBILE RADIO ASTRONOMY MOBILE-SATELLITE

BROADCASTING RADIO- BROADCASTING Radio- RADIONAVIGATION RADIOASTRONOMY ASTRONOMY location Earth) SATELLITE FIXED FIXED MOBILE FIXED FIXED- MOBILE

(space-to- MOBILE NAVIGATION NAVIGATION- MOBILE MOBILE MOBILE

Radio RADIOLOCATION MOBILE MOBILE SATELLITE Earth) MOBILE MOBILE RADIO- location RADIOLOCATION RADIOLOCATION

Standard Frequency (space-to-Earth) FIXED MOBILE- FIXED-SATELLITE FIXED FIXED FIXED FIXED FIXED MOBILE LOCATION MOBILE SATELLITE AMATEUR MOBILE

FIXED FIXED- and RADIO (passive) (passive) (space-to-Earth) MOBILE FIXED RADIO-

Space RADIO RADIO MOBILE SPACE Radio SATELLITE SPACE SPACE SPACE Time Signal (passive) (passive) MOBILE** RADIO SPACE RESEARCH SPACE

MOBILE- ASTRONOMY RADIO (passive) (passive) INTER- (passive) (space-to-Earth) EARTH FIXED INTER- SATELLITE RADIO SPACE RESEARCH SPACE RADIO RADIO NAVIGATION-

research (passive) SATELLITE RESEARCH (passive) RADIO ASTRONOMY RESEARCH FIXED-SATELLITE FIXED FIXED FIXED

SATELLITE FIXED- RESEARCH

RADIO- astronomy SPACE RESEARCH SPACE RESEARCH (space-to- FIXED

FIXED RADIO FIXED FIXED SATELLITE Satellite SATELLITE RADIO- SATELLITE (Earth-to-space) FIXED-SATELLITE FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE LOCATION ASTRONOMY

MOBILE FIXED FIXED

MOBILE RADIONAVIGATION-SATELLITE RADIO NAVIGATION MOBILE MOBILE FIXED MOBILE SATELLITE ASTRONOMY MOBILE MOBILE SPACE RESEARCH SPACE (deep space) FIXED MOBILE Space research RADIOLOCATION (active) RADIOLOCATION SATELLITE (deep space) (space-to-Earth) FIXED (space-to-Earth) ASTRONOMY FIXED ASTRONOMY SATELLITE EXPLORATION- MOBILE-SATELLITE (Earth-to-space) MOBILE-SATELLITE RADIONAVIGATION RADIOASTRONOMY Space research Space research (space-to-Earth) RADIONAVIGATION- This chart is a graphic single-point-in-time portrayal of the Table of Frequency Allocations used by the FCC and Radioastronomy NAVIGATION- SPACE RESEARCH (passive) SPACE (space-to-Earth) (Earth-to-space) ASTRONOMY FIXED (space-to-Earth) MOBILE FIXED ASTRONOMY Space research FIXED-SATELLITE

Earth) (space-to-Earth) RADIONAVIGATION- MOBILE ** RADIOLOCATION MOBILE (space-to-Earth) BROADCASTING SATELLITE RADIONAVIGATION RADIO ASTRONOMY MOBILE FIXED SPACE RESEARCH SPACE MOBILE MOBILE MOBILE-SATELLITE (Earth-to-space) MOBILE-SATELLITE SPACE RESEARCH SPACE Space research SATELLITE (passive) SATELLITE (space-to-Earth) NAVIGATION Radio astronomy

(space-to-Earth) (space-to-Earth) FIXED-SATELLITE (Earth-to-space) FIXED-SATELLITE EARTH EXPLORATION-SATELLITE (passive) EXPLORATION-SATELLITE EARTH NTIA. As such, it may not completely reflect all aspects, i.e. footnotes and recent changes made to the Table ASTRONOMY of Frequency Allocations. Therefore, for complete information, users should consult the Table to determine the current status of U.S. allocations. ISM - 61.25± 0.25 GHz ISM - 122.5± 0.500 GHz ISM - 245.0± 1 GHz U.S. DEPARTMENT OF COMMERCE 30GHz 300 GHz National Telecommunications and Information Administration * EXCEPT AERONAUTICAL MOBILE (R) Office of Spectrum Management ** EXCEPT AERONAUTICAL MOBILE

JANUARY 2016 PLEASE NOTE: THE SPACING ALLOTTED THE SERVICES IN THE SPECTRUM SEGMENTS SHOWN IS NOT PROPORTIONAL TO THE ACTUAL AMOUNT OF SPECTRUM OCCUPIED. For sale by the Superintendent of Documents, U.S. Government Printing Office Internet: bookstore.gpo.gov Phone toll free (866) 512-1800; Washington, DC area (202) 512-1800 Facsimile: (202) 512-2250 Mail: Stop SSOP, Washington, DC 20402-0001

Figure 1.4: Example of spectrum allocation, frequency range 0-300 GHz, United States, 2016. Applications include wireless communication, satellite communications, military and commercial radar, radio astronomy and license-free Industrial-Scientific-Medical (ISM) bands. More infor- mation can be found at: www.ntia.doc.gov and www.itu.int 8 CHAPTER 1. INTRODUCTION Chapter 2

System-level antenna parameters

2.1 Coordinate system and time-harmonic fields

The radiation properties of antennas are generally expressed in terms of spherical coordinates. The spherical coordinate system is shown in Figure 2.1. The unit vectors in this spherical coordinate system along the r, θ and φ directions are denoted by ~er, ~eθ and ~eφ, respectively. The location of p 2 2 2 point P (x, y, z) is indicated by the position vector ~r = x~ex + y~ey + z~ez = x + y + z ~er = |~r|~er.

Figure 2.1: Spherical coordinate system with dS = r2 sin θdθdφ = r2dΩ, where dΩ = sin θdθdφ is the solid angle.

In this book we will assume that all currents, voltages and electric and magnetic field components have a sinusoidal time variation. The instantaneous time-domain electric field E~ is now related to

9 10 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS a complex electric field E~ according to:

ωt E~(~r, t) = Re[E~ (~r)e ] = |E~ (~r)| cos (ωt + ϕ0), (2.1) where ϕ0 is the phase of the complex electric field E~ (~r), ω = 2πf is the angular frequency, f is the frequency of operation and Re[] denotes the real part of the complex variable. The electric field E~ (~r) in the space-frequency domain can now be written in terms of the spherical coordinates according to:

E~ (~r) = Er(~r)~er + Eθ(~r)~eθ + Eφ(~r)~eφ. (2.2)

2.2 Field regions

The main function of an antenna is to transform a guided electromagnetic (EM) wave along a transmission line in to radiated waves that propagate in free space. These radiated waves can, in turn, be received by another antenna. Due to reciprocity, the receiving antenna will transform the incident EM waves into a guided EM wave along a transmission line. One could also interpret this as the transition of electrons in conductors to photons in free space. The transition from transmission line to the antenna is illustrated in Fig. 2.2.

E-field direction at t=t0 ~

Generator Transmission line with a guided TEM wave

Antenna

Free-space spherical wave

Figure 2.2: Illustration of the transition of a guided wave along a transmission line to radiated spherical waves in free space using an antenna.

The electromagnetic field radiated by a transmitting antenna can be split up in three regions: 1) the near-field region, 2) transition or Fresnel region and 3) the far-field region. Fig. 2.3 shows an antenna and corresponding field regions. In the far-field region, the electric and magnetic field components are orthogonal to each other and orthogonal w.r.t. the direction of propagation, 2.3. FAR FIELD PROPERTIES 11

Near Field Fresnel Region Far Field Region

+ L -

Antenna

L3 2L2 R = 0.62 R = λ0 λ0

Figure 2.3: Field regions around an antenna. forming a so-called transverse electromagnetic (TEM) wave also known as plane wave. When the antenna is located in the origin of the coordinate system of Fig. 2.1, the transport of electromag- netic energy will be along the ~er direction, where ~er is the unit vector along the r-axis. Although there is not a very sharp transition between the various field regions, the most commonly used criterium to define the far-field region is the so-called Fraunhofer distance: 2L2 R = , (2.3) λ0 where R is the radial distance from the antenna, L is the largest dimension of the antenna and c λ = is the free-space wavelength of the radiated electromagnetic wave. Note that c = 3 · 108 0 f [m/s] is the speed of light.

2.3 Far field properties

Now let us position a transmit antenna in the origin of our coordinate system of Fig. 2.1 and a 2L2 receive antenna in the far-field region at a distance R > . In this case, the EM field at the λ0 receiving antenna will be locally flat, similar to an incident plane wave. A plane wave is a wave for which the equi-phase plane is perpendicular to the propagation direction. From undergraduate electromagnetic courses, it is well known that these plane waves in free space are transverse electromagnetic waves (TEM) with the following general form in the frequency domain:

−k0z (2.4) Ey(z) = Ae ,

ω where k0 = c is the free-space wavenumber and A the amplitude of the TEM wave. In (2.4) we have assumed that the plane wave propagates in the +z-direction. For the sake of simplicity 12 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS we only considered a component in the ~ey direction. The corresponding magnetic field H~ is perpendicular both to the E~ field and the direction of propagation. The transmit antenna will radiate spherical waves. In chapter 4 and 6 we will investigate the radiated fields in more detail. It will be shown that the far field in spherical coordinates E~ (~r) =

Eθ(~r)~eθ + Eφ(~r)~eφ + Er(~r)~er due to an antenna placed at the origin of our coordinate system at ~r = (0, 0, 0) can be expressed in the following form:

e−k0r E (~r) = E (θ, φ) , θ θ r e−k0r E (~r) = E (θ, φ) , (2.5) φ φ r

Er(~r) = 0.

Since EM waves in the far-field locally behave as TEM waves, we can write the corresponding magnetic field H~ in terms of the electric field:

1 H~ (~r) = ~er × E~ (~r), (2.6) Z0

q where Z = µ0 = 377Ω is the intrinsic free-space impedance. The radiated power density 0 0 expressed in Watts per square meter [W/m2], is known as the Poynting vector and describes the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. The Pointing vector in the far-field region only has a component in the radial direction ~ 2π ~er. The time-average Poynting vector Sp(~r) over a period T = ω at a specific position ~r in the far-field is now given by:

T 1 Z S~ (~r) = S~ (~r, t)dt p T p 0 T 1 Z = E~(~r, t) × H~ (~r, t)dt T 0 T 1 Z = Re[E~ (~r)eωt] × Re[H~ (~r)eωt]dt T 0 (2.7) T 1 Z 1 1 = (E~ (~r)eωt + E~ ∗(~r)e−ωt) × (H~ (~r)eωt + H~ ∗(~r)e−ωt)dt T 2 2 0 1 = [E~ (~r) × H~ ∗(~r) + E~ ∗(~r) × H~ (~r)] 4 1 = Re[E~ (~r) × H~ ∗(~r)]. 2

In the far-field region we can use relation (2.6). As a result, expression (2.7) can be written in 2.4. RADIATION PATTERN 13 the following form:

~ 1 ~ ~ ∗ 1 −1 ~ ~ ∗ Sp(~r) = 2 Re[E(~r) × H (~r)] = 2 Z0 Re[E(~r) × (~er × E (~r))]

1 −1 ~ ~ ∗ ~ ~ ∗ (2.8) = 2 Z0 Re[(E(~r) · E (~r))~er − (E(~r) · ~er)E (~r)]

1 −1 ~ 2 = 2 Z0 |E(~r)| ~er, since E~ (~r) · ~er = 0. We can conclude that the electromagnetic energy propagates in the radial direction ~er.

2.4 Radiation pattern

The radiation pattern describes the amount of power that is radiated in a certain direction in the far field of the antenna. It is a two- or three-dimensional graphical representation of the radiation properties of the antenna. From (2.5) we can observe that the electric field E~ has a r−1 dependence in the far-field region. The radiated power per element of solid angle dΩ = sinθdθdφ can be found from (see also Fig. 2.1):

2 P (~r) = P (θ, φ) = |r S~p(~r)|, (2.9) where Poynting’s vector is given by (2.8). Note that relation (2.9) is only valid in the far- field region in which the radiated power per element of solid angle is independent of r. In general, P (θ, φ) will have a maximum value. Let us assume that this maximum occurs at an angle (θ, φ) = (0, 0). The normalized radiation pattern F (θ, φ) is now defined as:

P (θ, φ) F (θ, φ) = . (2.10) P (0, 0)

Radiation patterns are always expressed in decibels [dB], that is using 10 log10(F (θ, φ)). Some typical examples are shown in Fig. 2.4 and Fig. 2.5. The main beam or main lobe of the antenna is directed towards the direction θ = 0. The other local maxima in the pattern are called sidelobes. The maximum sidelobe level is often one of the design parameters of an antenna, since the maximum sidelobe level determines the sensitivity of the antenna for (unwanted) interference from other directions. The height of the sidelobes is determined by a number of factors, including the size, shape and type of antenna. Some antennas, like small dipoles, do not have sidelobes: they only have a main lobe. In chapter 6 we will show in more detail how you can design array antennas with low sidelobes. From the normalized radiation pattern we cannot determine all quality measures of an antenna. Therefore, we will introduce some additional scalar antenna parameters, including beam width, directivity, antenna gain, effective aperture, aperture efficiency and input impedance. These parameters will be explained in more detail in the next sections. 14 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS

Radiation pattern of antenna with square uniform aperture A=16λ2, φ=0 plane 0 0

−5

−10

−15

−20

−25

−30

Antenna Radiation Pattern [dB] −35

−40

−45

−50 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 2.4: Two-dimensional normalized radiation pattern. A cut in the φ = 00-plane is shown. 2 The antenna has a uniformly-illuminated square-shaped aperture of size A = (4λ0) . The first sidelobe has a maximum level of -13.2 dB

3D radiation pattern of antenna with square uniform aperture A=16λ2 0 1

0.8

0.6

0.4

0.2 φ sin

θ 0 u=sin −0.2

−0.4

−0.6

−0.8

−1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 u=sin θ cos φ

Figure 2.5: Three-dimensional normalized radiation pattern of the antenna described in Fig.2.4 The indicated isolines correspond to [−40, −20, −3] dB level relative to the main lobe.

2.5 Beam width

The main beam of an antenna is characterized by the half-power beam width in the principle planes, i.e. the θ = 0 and φ = 0 plane. The beam width defines the beam area for which 2.6. DIRECTIVITY AND ANTENNA GAIN 15 the radiated or received power is larger than half of the maximum power. Therefore, one also often uses the term Half-Power Beam Width (HPBW). The beamwidth in the principle planes is expressed by θHP or as φHP .

2.6 Directivity and antenna gain

The directivity is a measure that describes the beamforming capabilities of the antenna, that is the concentration of radiated power in the main lobe w.r.t. other directions. Let us consider an antenna that radiates a total power of Pt Watts. When this antenna would be an isotropic radiator, the radiated power would be equally distributed over all directions with a corresponding P radiated power density of t per unit of solid angle. Note that an isotropic radiator cannot exist 4π in practise, it is only a theoretical reference that is used to define the directivity of a real antenna. The directivity function D(θ, φ) of a real antenna is now defined as the power density per unit of solid angle in the direction (θ, φ) relative to an isotropic radiator and is given by:

P (θ, φ) D(θ, φ) = . (2.11) Pt/4π The maximum of the directivity-function is the directivity D = max[D(θ, φ)]. All antennas have some form of losses, including Ohmic-losses in metal structures, losses in a supporting dielectric substrate and impedance-matching losses due to a mismatch between the antenna and transmission line (see also Fig. 2.2). Due to these losses not all the power that is provided by the source will be radiated. When we replace the total radiated power Pt in equation (2.11) by the total input power Pin that is provided to the antenna, we obtain the antenna gain function G(θ, φ), with:

P (θ, φ) G(θ, φ) = . (2.12) Pin/4π The maximum of the antenna gain function is called the antenna gain, G = max[G(θ, φ)]. The efficiency η of the antenna is now defined as: P η = t . (2.13) Pin This results in the following relation between the directivity and antenna gain:

G = ηD. (2.14)

2.7 Circuit representation of antennas

Typically, antennas are connected to a generator (transmit mode) or detector (receive mode) by means of a transmission line. More details on transmission line theory can be found in the next chapter. Sometimes antennas are directly connected to an amplifier, in which case power matching to a transmission line is not perse required. The input impedance of the antenna is an important measure that determines how much of the power which is transported along the 16 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS transmission line is actually radiated by the antenna. The input impedance of an antenna is in general complex, consisting of a resistive part Ra and a reactive part Xa according to:

Za = Ra + Xa (2.15)

= Rr + RL + Xa, where Rr represent the radiated losses and RL represent the real losses like Ohmic or dielectric losses. Note that the radiation resistance Rr is directly related to the total radiated power by the antenna and is in fact a measure that defines how well the antenna works. The reactance

Xa is related to the stored reactive energy in the close vicinity of the antenna. The antenna can p p now be represented by an one-port microwave network with input impedance Za = V /I , where V p is the voltage along the input ports of the antenna and Ip represents the current through the antenna. Fig. 2.6 shows the equivalent circuit of an antenna which is connected to a generator with internal impedance Zg = Rg. In practise, Rg = 50Ω will often be used. However, for optimal noise matching of the antenna to a low-noise amplifier (LNA) other (complex) values of the source impedance are preferred. We can define the radiation resistance Rr by considering the

Rg Ip + Antenna

+ + Vg Vg Vp Za= Ra+jXa - ~ -

-

Figure 2.6: Equivalent circuit representation of an antenna. The antenna is connected to a generator with internal impedance Rg.

p antenna in transmit mode. Let I be the current flowing through the antenna and Pt the total radiated power. The radiation resistance Rr is now defined as a resistance in which the power Pt is dissipated when the current Ip is flowing through the antenna. In other words:

P R = t . (2.16) r 1 p 2 2 |I | 2.8. EFFECTIVE ANTENNA APERTURE 17

The total radiated power Pt can be found by integrating the radiated power density over a sphere. Using relation (2.8) and (2.5), we obtain the following expression for Pt:

ZZ 2 Pt = S~p(~r) · ~err dΩ 2π π (2.17) 1 Z Z = Z−1 [|E (θ, φ)|2 + |E (θ, φ)|2] sin θdθdφ, 2 0 θ φ 0 0 At RF and microwave frequencies it is not possible to measure voltages and currents directly. As a consequence, we cannot measure the input impedance Za directly. However, at these frequencies it is possible to measure the (complex) amplitude of incident and reflected waves that travel along a transmission line. The input reflection coefficient of the antenna can be experimentally determined with a vector network analyser (VNA). Now let us assume that the antenna is connected to a t t transmission line with a characteristic impedance equal to Z0. Usually Z0 = 50Ω. Furthermore, we will assume that an incident transverse electromagnetic wave (TEM) wave propagates along the transmission line with complex amplitude a and a corresponding reflected wave (in the opposite direction) with complexe amplitude b. The reflection coefficient is now defined as the ratio between both complex amplitudes: b Γ = . (2.18) a From transmission line theory (see chapter 3 for more details), it is well known that the relation between the reflection coefficient and the input impedance at the input of the antenna is given by:

t Za − Z0 Γ = t . (2.19) Za + Z0

t In case the antenna is matched to the transmission line with Za = Z0, we will have zero reflection with Γ = 0.

2.8 Effective antenna aperture

Up to now, we have investigated the antenna parameters mainly by considering the antenna in transmit mode. The receive properties of the antenna are, of course, also very important. One of the key parameters that describes the antenna properties in receive mode is the effective antenna aperture Ae. Ae defines the equivalent surface in which the antenna absorbs the incident electromagnetic field that is dissipated by the load impedance ZL which is connected to the terminals of the antenna. 2 Now let Sp represent the power density (W/m ) of an incident plane wave. The effective antenna aperture is now defined as the ratio between the power Pr delivered to the load impedance and the power density Sp:

Pr Ae = . (2.20) Sp 18 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS

Where we have assumed that the antenna is optimally matched to the load impedance, according ∗ to Za = ZL. In chapter 4, we will introduce the reciprocity theorem. The reciprocity theorem defines the relation between the receive and transmit properties of an antenna. We will show that there is an unique relation between the antenna gain G and the effective antenna aperture Ae according to:

4πAe G = 2 . (2.21) λ0

2.9 Polarisation properties of antennas

Consider an antenna placed at the origin of our spherical coordinate system (see Fig. 2.1). The radiated electric field vector (in the far-field region) from the antenna usually will have two components, Eθ and Eφ, both perpendicular to the direction of propagation ~er. Both components (represented in the frequency domain) will usually have a relative phase difference with respect to each other. When the phase difference is equal to 0, the resulting field will be linearly polarized, since both components can simply be added in a vectorial way. So the direction of the electric

field vector in the time domain is constant. In case of a phase difference between Eθ and Eφ, the direction of the resulting time-domain electric field E~(~r, t) will vary over time with a period of T = 2π/ω. The time-domain electric field vector will now describe an ellipse. Therefore, this situation is called elliptical polarisation. Let us investigate this situation in more detail. At far-field distances from the antenna, we can assume that the fields are locally flat and can be represented by a plane wave. Now assume that we have a plane wave propagating in the ~er direction. The time-domain electric field can now be written as:

Eθ(~r, t) = E1(~r) cos (ωt − k0r), (2.22)

Eφ(~r, t) = E2(~r) cos (ωt − k0r + ϕ), where ϕ is the phase difference between both electric-field components Eθ and Eφ (frequency- domain). To reduce the complexity in our notation, we will omit the place-time dependence (~r, t) in the rest of this section. In addition, we will choose r = 0. In order to determine the equation that describes the elliptical trajectory of the resulting electric field vector E = (Eθ, Eφ), we need to eliminate the time-dependence. This can be done by writing Eφ in the following form:

Eφ = E2(cos ωt cos ϕ − sin ωt sin ϕ) (2.23) Eθ = E2( cos ϕ − sin ωt sin ϕ), E1 resulting in: !  E E 2    E 2 φ − θ cos ϕ = sin2 ωt sin2 ϕ = 1 − cos2 ωt sin2 ϕ = 1 − θ sin2 ϕ. (2.24) E2 E1 E1 2.9. POLARISATION PROPERTIES OF ANTENNAS 19

This can be rewritten in the following form:

 E 2  E 2 2E E θ + φ − θ φ cos ϕ = sin2 ϕ. (2.25) E1 E2 E1E2 It can be shown that the resulting equation describes an ellipse in the plane perpendicular to the direction of propagation ~er. The orientation of the major and minor axis of the ellipse usually do not coincide with the unit vectors ~eθ and ~eφ. The ratio between the major and minor axis of the ellipse is the so-called axial ratio (AR), with AR ≥ 1. Fig. 2.7 illustrates the polarization ellipse and orientation of the major and minor axis of the ellipse.

Figure 2.7: Polarization ellipse (a function of time) and the polarization pattern (a function of ‘ |E1| space) [9]-[10]. Note that the axial ratio is determined by AR = ‘ ≥ 1. We have assumed in |E2| this figure that ~eθ coincides with the x direction and ~eφ coincides with the y direction

The axial ratio is an important quantity to describe the quality of circularly-polarized antennas. Now let is investigate this special case in more detail.

i. Circular polarisation (E1 = E2 = E and ϕ = ±π/2). Then (2.25) can be written as:

E 2 E 2 θ + φ = 1. (2.26) E E

This represents the equation of a circle. Therefore, we now have a circularly-polarized field. We can distinguish two particular situations, the first one with ϕ = π/2:

E~L = E (~eθ cos ωt − ~eφ sin ωt) , (2.27) 20 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS

and in case ϕ = −π/2, we obtain:

E~R = E (~eθ cos ωt + ~eφ sin ωt) . (2.28)

In the first situation, the E~-vector rotates anti clock-wise w.r.t. the direction of propagation. Therefore, this case is called Left-Hand Circular Polarisation (LHCP). In the second situation (2.28), the E~-vector rotates clock-wise, corresponding to Right-Hand Circular Polarisation (RHCP). In a practical situation, antennas will never be perfectly circularly polarized. We

will always have an E~L als de E~R field component. De axial ratio describes the quality of circular polarization and can be expressed in term of the frequency-domain field components

E~L en E~R according to:

|EL| + |ER| AR = . (2.29) |EL| − |ER|

ωt where E~L = Re[E~Le ]. When AR = 1, we would have a perfectly circularly-polarized wave.

ii. Linear polarisation (E1 6= E2 en ϕ = ±π). Equation (2.25) can now be written as: E E θ + φ = 0, (2.30) E1 E2 which mathematically represents a straight line. In this case we obtain linear polarization, where AR = ∞.

2.10 Link budget analysis: basic radio- and radar equation

The antenna is one of the components of a complete system. Examples of such systems include wireless communication systems (e.g. 4G/5G) and radar. The antenna parameters that we have introduced in the previous sections can be used to quantify the quality of a radio link or to determine the range of a radar. Fig. 2.8 shows a schematic illustration of a wireless link, consisting of a transmit and receive antenna. The transmit antenna is connected to a transmitter that can generate Pt Watts of power. We will assume that both antennas have the same polarisation and orientation with respect to each other. The power density in (W/m2) at the receive antenna is now given by: P G S = t t , (2.31) r 4πr2 where Gt is the antenna gain of the transmit antenna as compared to an isotropic antenna. The power received by the receive antenna with an effective aperture Aer now becomes: P G A P = t t er . (2.32) r 4πr2 λ2G This equation is also known as the radio equation or Friis equation. By using A = 0 r in(2.32), er 4π we finally obtain: P G G λ2 P = t t r 0 , (2.33) r (4π)2r2 2.10. LINK BUDGET ANALYSIS: BASIC RADIO- AND RADAR EQUATION 21

Transmit antenna Receive antenna Gt Gr , Aer

Transmitter Receiver Pt Pr

r

Figure 2.8: Radio link between a transmit antenna (Tx) and a receive antenna (Rx) over a distance r.

! λ2 where G is the antenna gain of the receiving antenna. The term 0 is also known as the r (4π)2r2 free-space path loss and depends on frequency, since λ0 = c/f. However, this term is not really a loss, since we had assumed in our ideal radio link of Fig. 2.8 that the entire system, including free-space, is lossless. The frequency dependence comes from the relation between the antenna gain Gr and the effective area Aer. It tells us that for a constant antenna gain, the effective area scales with 1/f 2. This implies that at higher frequencies antennas with a much larger antenna gain need to be used in order to maintain the same range as compared to lower frequencies. Note that a large antenna gain also implies a more directive antenna with a narrow beam. This complicates the implementation of omni-directional wireless communication at millimeter-wave frequencies. Phased-arrays with electronic beam scanning offer a solution for this problem, see chapter 6.

As an example of a link-budget analysis, consider the Bluetooth system operating in the 2.4−2.48

GHz band. Assume omni-directional antennas (Gt = Gr = 1), an output power of Pt = 1 mW (0 −10 dBm) and a receiver sensitivity Pr,min = 10 W (= −70 dBm). The maximum range rmax of this system then becomes:

s 2 PtGtGrλ0 rmax = 2 ≈ 30 [m]. (2.34) (4π) Pr,min Let us now consider a radar system. In this case, we replace the receive antenna by an object with a radar cross section σ[m2]. The radar cross section of an object depends on the size and shape (e.g. car or airplane) and determines how much of the incident power is reflected back to the radar. We will assume that the transmit antenna can also be used as receive antenna, therefore 22 CHAPTER 2. SYSTEM-LEVEL ANTENNA PARAMETERS

Gt = Gr = G. The latter is due to the reciprocity theorem which applies to passive microwave structures including antennas. The received power of the radar is now given by: P G σ P G2σλ2 P = t t A = t 0 . (2.35) r 4πr2 4πr2 er (4π)3r4 This equation is known as the radar equation. It is a first-order estimation of the received power and shows that the probability of detection of an object decreases according to r4. In a real system, the radar equation will be somewhat more complicated. In addition, the maximum detection range can be increased by using advanced signal processing techniques. More background information can be found in Skolnik [11]. Now assume that the receiver of the radar requires a minimum power of Pmin to detect an object. The maximum radar range rmax is now given by:

" 2 2 #1/4 PtG σλ0 rmax = 3 . (2.36) (4π) Pmin

2 As an example, consider an X-band radar operating at f = 10 GHz with Ae = 1m , Pt = 10 2 −13 kW, σ = 1m en Pmin = 10 W (10 log10 Pmin = −100 dBm). This implies that G = 14000 (GdB = 41 dB). The maximum range now becomes rmax = 54 km. Chapter 3

Transmission line theory and microwave circuits

3.1 Introduction

In this chapter we will extend the well-known circuit theory to transmission line theory. Circuit theory can be applied to electrical circuits of which the size of the individual (lumped) components, like resistors, capacitors and inductors, are much smaller than the electrical wavelength (size  0.1λ). In case the physical dimension of a component or network becomes a significant fraction of the wavelength (size > 0.1λ), we need to apply transmission line theory. We consider the transmission line to be a distributed network on which the voltages and currents vary along the location along the line. In this chapter, we will derive the transmission line concepts by using circuit theory concepts. Therefore, we do not need to solve Maxwell’s equations explicitly to analyse networks based on transmission lines.

3.2 Telegrapher equation

Fig. 3.1 illustrates some examples of transmission lines. Well-known transmission lines used in many microwave applications are the coaxial cable, microstrip lines and waveguides. In addition, long range 3-phase electrical power lines operating at 50 Hz should also be considered as trans- mission lines. From basic electromagnetics courses, it is known that a transverse electromagnetic wave (TEM) with zero cut-off frequency can propagate along lines consisting of at least two metal conductors, e.g. coaxial cable. These type of transmission lines also allow other modes, e.g. transverse electric (TE) or transverse magnetic (TM), to propagate above their cut-off frequen- cies. However, in this chapter we will assume that only a TEM mode can propagate along the transmission line. A plane wave is also an example of a TEM wave, see section 2. Note that along a waveguide (see Fig. 3.1) a TEM mode cannot exist, only TE and TM modes can propagate. However, the theory introduced in this chapter can also be applied to waveguides, as long as only a single mode can propagate. Now let us consider a general transmission line that carries a TEM wave that propagates along the +z-direction, as illustrated in Fig. 3.2. The transmission line is connected to a source. As a

23 24 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

GND

Coaxial cable Parallel wires (twin lead)

Microstrip lines

GND

Strip lines

GND

Microstrip lines

Multi-layer Printed Circuit Board (PCB) High-voltage lines

Figure 3.1: Examples of transmission lines used in various applications. result of this, a voltage v(z, t) and current i(z, t) will exist at the location z.

i(z,t)

+ v(z,t)

-

∆z z

Figure 3.2: Two-wire representation of a transmission line. A TEM wave propagates in the +z-direction.

We would like to apply Kirchhoff’s voltage and current laws instead of Maxwells’ equations to analyse the characteristics of the transmission line. This can be done by considering a small section between z and z + ∆z of the transmission line, as illustrated in Fig. 3.2. Since this small section is electrically very small (∆z  λ), we can model this small section using lumped-element 3.2. TELEGRAPHER EQUATION 25 circuit components as illustrated in Fig. 3.3, where the lumped element are defined per unit length:

L = series inductance per unit length [H/m],

C = shunt capacitance per unit length [F/m], (3.1) R = series resistance per unit length [Ω/m],

G = shunt conductance per unit length [S/m].

i(z,t)

+ v(z,t)

- z ∆z

i(z,t) i(z+∆z,t)

+ ∆ + R∆z L z ) ∆ ) v(z,t G∆z C∆z v(z+ z,t - -

∆z

Figure 3.3: Lumped element circuit representation of a small section ∆z of a transmission line.

The series inductance L∆z originates from the magnetic field that is induced by the current i(z, t) flowing on the line and the series resistance R∆z accounts for the related losses in the metal conductor. The shunt capacitor C∆z is created by the potential difference v(z, t) which creates an electric field between the two wires. Dielectric losses related to the dielectric material between the two wires are represented by the shunt conductance G∆z. We can now apply Kirchhoff’s voltage and current laws to the equivalent circuit of a small section ∆z. Kirchhoff’s voltage law provides:

∂i(z, t) v(z, t) − R∆zi(z, t) − L∆z − v(z + ∆z, t) = 0. (3.2) ∂t Similarly, Kirchhoff’s current law gives:

∂v(z + ∆z, t) i(z, t) − G∆zv(z + ∆z, t) − C∆z − i(z + ∆z, t) = 0. (3.3) ∂t 26 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

Now divide (3.2) and (3.3) by ∆z and take the limit of ∆z → 0, resulting in two differential equations with two unknowns (v(z, t) and i(z, t)) which can be written in the following form:

∂v(z, t) ∂i(z, t) = −Ri(z, t) − L , ∂z ∂t (3.4) ∂i(z, t) ∂v(z, t) = −Gv(z, t) − C . ∂z ∂t

Equation (3.4) is known as the Telegrapher equation in the time domain. Since we assume a sinusoidal time variation of all field components in this book, the voltage and current can be written as:

v(z, t) = Re[V (z)eωt], (3.5) i(z, t) = Re[I(z)eωt], in which ω = 2πf and f is the frequency of operation. In this case the time derivative of v(z, t) in (3.4) transform into a simple multiplication in the frequency domain:

∂v(z, t) ↔ ωV (z). (3.6) ∂t By applying this in (3.4) we obtain the frequency-domain Telegrapher equation:

∂V (z) = − (R + ωL) I(z), ∂z (3.7) ∂I(z) = − (G + ωC) V (z). ∂z

The general solution of the frequency-domain Telegrapher equation is now a combination of a voltage and current wave propagating the +z direction and a wave propagating in the −z direction:

+ −γz − γz V (z) = V0 e + V0 e , (3.8) + −γz − γz I(z) = I0 e + I0 e , where γ is the complex propagation constant:

q γ = α + β = (R + ωL)(G + ωC), (3.9) in which α is the attenuation of the line and β is phase constant. The term e−γz corresponds to a wave propagating in the +z direction and eγz to a wave propagating in the −z direction. Note that V (z) and I(z) in (3.8) represent the total voltage and current, respectively, at the position z along the line. It can be easily verified that (3.8) describes the general solution by substituting 3.3. THE LOSSLESS TRANSMISSION LINE 27

(3.8) in to the Telegrapher equation (3.7). By doing this, we can also obtain a relation between the complex voltage and current amplitudes:

+ γ + I0 = V0 , R + ωL (3.10) −γ I− = V −. 0 R + ωL 0

In addition, the characteristic impedance Z0 of the transmission line is defined by:

+ V0 R + ωL Z0 = + = . (3.11) I0 γ The time-domain representation of the voltage along the transmission line can be obtained by using the frequency domain solution (3.8) and relation (3.5), resulting in:

+ +
−αz − −
−αz (3.12) v(z, t) = |V0 |cos ωt − βz + φ e + |V0 |cos ωt − βz + φ e , where it can be observed that α describes the exponential attenuation along the line and where the phase constant β determines the wavelength on the line:

2π λ = . (3.13) β

− Question: Why is there a − sign in the expression of I0 in (3.10)?

3.3 The lossless transmission line

In the previous section we have derived the solution for the voltage and current along a trans- mission line with losses, resulting in a complex propagation constant γ = α + β and complex characteristic impedance Z0. In a lot of practical applications, the losses along the transmission line are very small and can, therefore, be neglected for a first-order analysis. When the trans- mission line is lossless, the series resistance R and the shunt conductance G in Fig. 3.3 can be neglected. Therefore, the propagation constant and characteristic impedance now become real:

√ γ = β = ω LC, √ β = ω LC, (3.14) α = 0, s L Z = . 0 C 28 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

The corresponding wavelength and associated phase velocity of the wave along the line are given by:

2π 2π λ = = √ β ω LC (3.15) ω 1 vp = = √ . β LC Example A transmission line operating at f = 500 MHz has the following per unit length parameters:

L = 0.2[µH/m],

C = 300[pF/m], (3.16) R = 5[Ω/m],

G = 0.01[S/m].

Using (3.9), we find that the complex propagation coefficient is calculated by: q γ = α + β = (R + ωL)(G + ωC) = p−590 − 10.99 = 0.23 + 24.3 [1/m], (3.17) in which the square root was evaluated according to Im(γ) > 0. Re-calculating this for a lossless line with R = G = 0 gives: √ γ = β = ω LC = 24.3 [1/m]. (3.18)

3.4 The terminated lossless transmission line

Consider the terminated lossless transmission of Fig. 3.4. The line is terminated at z = 0 with a complex load impedance ZL = RL + XL. The total voltage and current at the position z along the line are found by (3.8) and (3.10) and using γ = β:

+ −βz − βz V (z) = V0 e + V0 e , (3.19) V + V − I(z) = 0 e−βz − 0 eβz. Z0 Z0 + − This equation still has two unknown coefficients V0 and V0 . A relation between both coefficients can be found by using the relation between the voltage and current along the load impedance. At z = 0 this relation is given by:

+ − VL V (0) V0 + V0 (3.20) ZL = = = + − Z0. IL I(0) V0 − V0 3.4. THE TERMINATED LOSSLESS TRANSMISSION LINE 29

V(z), I(z) IL

+

Z 0, β VL ZL -

z -l 0

Figure 3.4: The terminated lossless transmission line, terminated with a complex load impedance

ZL

Rewriting (3.20) gives:

− ZL − Z0 + (3.21) V0 = = V0 . ZL + Z0

We can now introduce the reflection coefficient Γ as:

− V0 ZL − Z0 (3.22) Γ = = + = . V0 ZL + Z0

Some special terminations are:

ZL = Z0, Γ = 0, matched load ,

(3.23) ZL = 0, Γ = −1, short circuit termination,

ZL = ∞, Γ = 1, open circuit termination,

In case of a matched load ZL = Z0, the voltage along the line is constant |V (z)| = |V0|. This line is now also called to be ”flat”. In other cases when Γ = |Γ|eθ 6= 0, the magnitude of the voltage will not be constant over the line:

+ 2βz + (θ−2βl) (3.24) |V (z)| = = V0 1 + Γe = V0 1 + |Γ| e , at z = −l. 30 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

The voltage along the line fluctuates according to:

+ Vmax = V0 |1 + |Γ|| ,

+ (3.25) Vmin = V0 |1 − |Γ|| , V 1 + |Γ| VSWR = max = , Vmin 1 − |Γ| where VWSR is the voltage standing wave ratio which is quantity that describes the fluctuation of the voltage along the line. The VSWR is an important quantity that is often used to specify the load conditions of active devices, like power amplifiers (PAs). PAs use semiconductor transistors which have a maximum break-down voltage. By specifying a maximum VWSR, it is avoided that the maximum voltage along the line exceeds the break-down voltage of the transistor. The reflection coefficient as defined in (3.22) can be generalized to any point z = −l on the transmission line according to:

V −e−βl Γ(z = −l) = 0 = Γe−2βl, (3.26) + βl V0 e where Γ = Γ(z = 0). Now let us consider the configuration of Fig. 3.5. The terminated line with characteristic impedance Z1 is now connected to another (very long) transmission line with characteristic impedance Z0. The input impedance Zin looking into the terminated line from the location z = −l is now given by:

+ h βl −βli V (z = −l) V0 e + Γe Z = = Z in I(z = −l) + βl −βl 0 V0 [e − Γe ] (3.27) ZL + Z0 tan βl = Z0 . Z0 + ZL tan βl Equation (3.27) shows that the input impedance of a terminated line can be tuned to a specific value by changing the length l of the transmission line. This can be very useful in matching circuits and filters, as we will see lateron. Example

Consider a lossless transmission line of length l which is short-circuited at the termination (ZL = 0). As expected in case of a short-circuit, we find from (3.22) that the reflection at the termination Γ = −1. The voltage and current along the line is determined from (3.19) and takes the form:

+ V (z) = −2V0 sin βz, 2V + I(z) = 0 cos βz, (3.28) Z0

Zin(z = −l) = Z0 tan βl. 3.4. THE TERMINATED LOSSLESS TRANSMISSION LINE 31

ZL Z 0 Z 1

z −l 0

Z in

Figure 3.5: The terminated lossless transmission line with characteristic impedance Z1 connected at z = −l to another transmission line with characteristic impedance Z0.

The corresponding VSWR = ∞ since the minimum voltage on the line Vmin = 0. When a power amplifier is connected to a short circuited transmission line, the output voltage at the output port of the amplifier may become large than the breakdown voltage. In this case, the amplifier will be destroyed.

When the terminated transmission line is not well matched to the connecting transmision line, that is Zin 6= Z0, part of the incident power will be reflected. Now let us assume that the load impedance is purely resistive ZL = RL. The incident power Pin and reflected power Pr can be related by using the magnitude of the voltage reflection coefficient: 2 |Vr| Pr 2RL 2 = 2 = |Γ| . (3.29) Pin |Vin| 2RL Note that in (3.29) we have used the time-average power which can be calculated as:

T 1 Z P (z) = P(z, t)dt av T 0 T 1 Z = v(z, t)i(z, t)dt (3.30) T 0 1 = Re [V (z)I∗(z)] , 2 in which a similar derivation as used in (2.7) was applied. Note that the time domain voltage and current can be expressed as:

h i h i v(z, t) = Re V (z)eωt = Re |V (z)|eωt+φv (3.31) h i h i i(z, t) = Re I(z)eωt = Re |I(z)|eωt+φi . 32 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

Exercise. Derive the expression for time-average power, given by (3.30), by using relation (3.31).

The return loss RL is now defined as the amount of power reflected by the load:

2 (3.32) RL = −10 log10 |Γ| = −20 log10 |Γ|.

Related to return loss is the insertion loss. The insertion loss describes the loss of a transition between two transmission lines, as illustrated in Fig. 3.6. The transmitted wave for z > 0 takes according to (3.19) the form:

+ −βz V1(z) = V0 T e , (3.33) where T is the transmission coefficient: Z − Z 2Z T = 1 + Γ = 1 + 1 0 = 1 . (3.34) Z1 + Z0 Z1 + Z0

Let Pin be the power of the incident wave and P1 the power of the transmitted wave. The insertion loss IL is now found as:  |V |2    1 P1  2Z1  IL = −10 log10 = −10 log10  + 2  Pin  |V |  0 (3.35) 2Z0   Z1 = −20 log10 |T | + 10 log10 . Z0

Z 0 Z 1

z 0

Figure 3.6: Interface between two transmission lines causing insertion loss.

3.5 The quarter-wave transformer

The very nice application of the terminated transmission line is the so-called λ/4 transformer as illustrated in Fig. 3.7. A transmission line section of length l = λ/4 with a characteristic impedance Z1 is terminated by a load impedance RL. In this section we will only consider 3.5. THE QUARTER-WAVE TRANSFORMER 33 a resistive load. The transformer is connected to another transmission line with characteristic impedance Z0 of which we will assume that the length is infinitely long or connected to a matched source with impedance Zs = Z0 (no reflections from the source). The impedance looking into

R Z 0 Z 1 L

z −λ/4 0

Z in

Figure 3.7: The λ/4 transformer in which a λ/4-transmission line is connected to a terminated lossless transmission line, terminated with a load impedance RL the transformer at z = −λ/4 can be calculated using (3.27), where it should be noted that the quarterwave line has a characteristic impedance Z1. Using l = λ0/4 and ZL = RL we get:

RL + Z1 tan βl Zin = Z1 Z1 + RL tan βl RL + Z1 tan βλ/4 (3.36) = Z1 Z1 + RL tan βλ/4 Z2 = 1 , RL

 2π   λ  π where βl = λ 4 = 2 . From (3.36) we find that the reflection coefficient at z = −λ/4 is zero when the following condition is satisfied:

Zin − Z0 p (3.37) Γ(z = −λ/4) = = 0 −→ Zin = Z0 −→ Z1 = Z0RL. Zin + Z0

Example

Consider a load resistance RL = 100 Ω which needs to be matched to a long transmission line with characteristic impedance Z0 = 50 Ω. Design a quarter-wave transformer at the frequency f0 and plot the magnitude of the input reflection coefficient versus the normalized frequency f/f . From √ 0 (3.37) we find that Z1 = Z0RL = 70.71 Ω. The magnitude of the input reflection coefficient is 34 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS found from:

Zin − Z0 |Γ| = , Zin + Z0 RL + Z1 tan βl (3.38) Zin = Z1 Z1 + RL tan βl πf βl= . 2f0 Fig. 3.8 shows a plot of the amplitude of the reflection coefficient of this quarter-wave transformer. We can observe that only at the design frequency an optimal matching is obtained. This is a clear drawback of a single-stage quarter-wave transformer and limits the frequency bandwidth of such a matching circuit. The bandwidth can be increased by using multiple stages. Note that the transformer is also matched at l = nλ/4 with n = 3, 5, .....

0.35

0.3

0.25

0.2

0.15 abs(Gamma)

0.1

0.05

0 0 0.5 1 1.5 2 2.5 3 3.5 4 f/f0

Figure 3.8: Magnitude of the input reflection coefficient versus the normalized frequency f/f0 of a λ/4 transformer terminated with a load impedance RL = 100 Ω

3.6 The lossy terminated transmission line

In practise, transmission lines will always be lossy, although the theory of loss-less lines is generally a very good first-order estimation. In case of a lossy terminated line, as illustrated in Fig. 3.9, the voltage and current along the line are found using the general solution of the Telegrapher equation (3.8) and can be written in the following form:

+ −γz γz
V (z) = V0 e + Γe , (3.39) V + I(z) = 0 e−γz − Γeγz
, Z0 3.6. THE LOSSY TERMINATED TRANSMISSION LINE 35 where Γ is the reflection coefficient at the termination z = 0. The input impedance at z = −l is now found by:

+ h γl −γli V (z = −l) V0 e + Γe Z = = Z in I(z = −l) + γl −γl 0 V0 [e − Γe ] (3.40) ZL + Z0 tanh γl = Z0 . Z0 + ZL tanh γl

V(z), I(z) IL

+

Z 0, α, β VL ZL -

z -l 0

Figure 3.9: The lossy terminated transmission line with γ = α + β.

The power delivered at z = −l is now:

1 P = Re (V (−l)I∗(−l)) in 2 2 (3.41) + V0   = e2αl − |Γ|2e−2αl . 2Z0

The corresponding power delivered to the load at z = 0 is found by:

1 P = Re (V (0)I∗(0)) L 2 2 (3.42) + V0   = 1 − |Γ|2 . 2Z0

The loss in the terminated line of length l is found as the difference between Pin and PL:

2 V + 0 h 2αl i 2 h −2αli (3.43) Ploss = Pin − PL = e − 1 + |Γ| 1 − e . 2Z0

The first term in (3.43) is the loss of the incident wave along the line with length l. The second term describes the loss of the reflected wave. 36 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

3.7 Field analysis of transmission lines

Real transmission lines may take various shapes. Fig. 3.10 shows the cross section of various types of transmission lines that support TEM or quasi-TEM wave propagation. The physical pa- rameters like width of a microstrip line and dielectric constant and loss tangent of the substrate can be translated into transmission parameters Z0, α, β and/or in terms of L, C, R, G. However, for most transmission line types this cannot be done in an analytical way, since we need to solve Maxwells’ equation using the boundary conditions on the metal structures. Approximate models can be applied, but numerical methods need to be used in order to obtain very accurate results. Nowadays, several open-source and commercial software packages, like QUCS and ADS are avail- able with tools to determine the transmission line parameters for various types of transmission lines. A type of transmission line for which we can easily solve Maxwells’ equations is the parallel-

GND

Coaxial cable, TEM mode

Parallel wires (twin lead), TEM mode

GND

Stripline, TEM mode

GND

Microstrip, quasi-TEM mode

GND

GND GND

Co-planar waveguide, quasi-TEM mode

GND

Metal waveguide, non-TEM mode GND

Figure 3.10: Cross sections of various transmission lines that support TEM, quasi-TEM or non- TEM wave propagation. plate waveguide as illustrated in Fig. 3.11. Although it is not a very practical transmission line, its characteristics are quite similar to more practical transmission lines, e.g. microstrip line. It consists of two large metallic plates separated by a dielectric substrate with permittivity  = r0 and permeability µ0. We will assume that the transmission line is lossless. In addition, it is assumed that the width w  d. The TEM wave has zero cut-off frequency and propagates along −βz the z direction, therefore, Ez = Hz = 0 and E~ = E~t = (Ex~ex + Ey~ey)e . In this case, it can be 3.7. FIELD ANALYSIS OF TRANSMISSION LINES 37

y Φ(y=d)=V0

Φ(y=0)=0

ε,µ0 d x w

z

Figure 3.11: The parallel-plate transmission line that support a TEM wave shown that the transverse components of the electric and magnetic field can be found by solving Laplace’s equation:

2 ~ ∇t Et = 0, (3.44) 2 ~ ∇t Ht = 0,

~ 2 2 2 2 2 where the transverse electric field is Et = Ex~ex + Ey~ey and ∇t = ∂ /∂x + ∂ /∂y resembles the Laplacian operator in the (x, y) dimensions. From (3.44) we can observe that the transverse fields are the same as the static fields in a parallel plate configuration. From electrostatics, we know that the electric field in (3.44) can be expressed as a gradient of a scalar potential Φ(x, y):

(3.45) E~t = −∇tΦ(x, y),

This scalar potential Φ(x, y) satisfies Laplace’s equation:

2 (3.46) ∇t Φ(x, y) = 0.

The boundary conditions of this differential equation are as indicated in Fig. 3.11:

Φ(x, 0) = 0, (3.47)

Φ(x, d) = V0.

A solution of Φ(x, y) cannot depend on x, since we have assumed that the width w is very large (close to infinite). Therefore, Φ(x, y) takes the form:

V y Φ(x, y) = 0 , (3.48) d 38 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS which can be easily verified by substituting this in (3.46) and (3.47). The electric and magnetic fields between the fields are now obtained using (3.45):

−βz V0 −βz E~ (x, y, z) = −∇tΦ(x, y)e = −~ey e , d (3.49) 1 V H~ (x, y, z) = ~e × E~ (x, y, z) = ~e 0 e−βz, η z x ηd √ p where the propagation constant β = ω µ0 and the intrinsic impedance η = µ0/. The corre- sponding voltage and current along the line are now given by:

Z d −βz V (z) = − Eydy = V0e 0 (3.50) Z w Z w Z w wV0 −βz I(z) = Jzdx = (−~ey × H~ ) · ~ezdx = Hxdx = e . 0 0 0 ηd The transmission line parameters are now found by (with R = G = 0):

V ηd Z = = [Ω], 0 I w ZZ   µ0 ~ 2 µ0d H L = 2 |H| dS = , |I| w m (3.51) ZZ    ~ 2 w F C = 2 |E| dS = , |V0| d m √ √  1  β = ω LC = ω µ . 0 m

3.8 The Smith chart

The Smith chart is a very usefull graphical representation to visualize the behaviour of transmission- line circuits. The Smith chart was developed by P. Smith in 1939, well before computer-based design tools have become available. The smith chart is still de-facto de standard representation tool in microwave engineering. The Smith chart is basically built around the complex represen- tation of the reflection coefficient of a transmission line circuit, for example as described in (3.37) in case of a quarter-wave transformer. The reflection coefficient is complex and can be expressed in terms of amplitude and phase:

Z − Z in 0 φΓ (3.52) Γ = = Γr + Γi = |Γ|e . Zin + Z0 When we would plot Γ in the complex plane, we find that all values are within the unit circle, since

|Γ| ≤ 1. In case Zin = Z0, Γ = 0 and we would end up in the centre of the unit circle. The most common form of the Smith chart is shown in Fig. 3.12. Note that the normalized impedance values zin = rin +xin = Zin/Z0 are listed in the Smith chart, where rin, xin are the normalized resistance and reactance, respectively. Let us now investigate some special cases in more detail. First assume 3.8. THE SMITH CHART 39

Figure 3.12: The Smith chart. The normalized impedance values zin = rin + xin = Zin/Z0 are listed. Usually Z0 = 50 Ω.

that are transmission line circuit is purely resistive with Zin = Rin + Xin = Rin = rinZ0. In this case we find that:

Z − Z r − 1 Γ = in 0 = in . (3.53) Zin + Z0 rin + 1

Now Γ is real and describe a line along the horizontal axis with Γ = −1 for rin = 0 (short),Γ = 0 for rin = Z0 (matched load) and Γ = 1 for rin = ∞ (open). Usually, Z0 = 50 Ω is used. Another special case occurs when the load is purely reactive with Zin = Xin = xinZ0. It can be shown that the reflection coefficient Γ now describes a circle with amplitude |Γ| = 1. This can be generalized to the case of a constant resistance and varying reactance. In this case we obtain circles as can be observed in Fig. 3.12. The lines for a load with constant reactance and varying resistance can also be observed in the Smith chart and correspond to parts of large circles that extend the unit circle. Smith charts with both impedance and admittance values are also available and are useful when designing serial or parallel matching circuits. Fig. 3.13 shows such a chart. Exercise. Plot the following load conditions in the Smith chart: 40 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

Figure 3.13: Smith chart with normalized impedance and admittance coordinates.

i. Zin = 0,

ii. Zin = ∞,

iii. Zin = 50 Ω,

iv. Zin = 50 − 150 Ω.

3.9 Microwave networks

Consider a microwave network with K-ports as illustrated in Fig. 3.14. Each port is connected to a transmission line on which TEM-modes can propagate. At port k, the total voltage and current is given by:

+ − Vk = Vk + Vk , (3.54) + − Ik = Ik − Ik . 3.9. MICROWAVE NETWORKS 41

Microwave network

I1 Ik IK + - + - + - V1 Vk VK

Figure 3.14: K-port microwave network with port voltages and currents.

The relation between the port voltages and port current is defined by the impedance matrix [Z]:        V   Z Z ..Z   I   1   11 12 1K   1                           V2   Z21 Z22 ..Z2K   I2          =     . (3.55)              .   .....   .                          VK ZK1 ZK2 ..ZKK IK

In most practical case we will investigate two-port microwave networks with K = 2. The relation between port voltages and currents then reduces to:      

 V1   Z11 Z12   I1          =     , (3.56)             V2 Z21 Z22 I2 where the individual components of the Z-matrix are found by:

V1 V1 Z11 = Z12 = I I 1 I2=0 2 I1=0 (3.57)

V2 V2 Z21 = Z22 = I1 I2=0 I2 I1=0 Example Consider the network of Fig. 3.15, where a 50 Ω resistor is connected in parallel. The components of the impedance matrix are now found using (3.57):

V1 V1 Z11 = = 50 Ω Z12 = = 50 Ω I I 1 I2=0 2 I1=0 (3.58)

V2 V2 Z21 = = 50 Ω Z22 = = 50 Ω I1 I2=0 I2 I1=0 42 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

I1 I2 → ← + +

V1 50 Ω V2 − −

Figure 3.15: 2-port microwave network with a parallel 50 Ω resistive load.

The voltage-current relation of a microwave network can also be expressed by means of an admit- tance matrix [Y ]. In case of a two-port network, we obtain:

     

 I1   Y11 Y12   V1          =     , (3.59)             I2 Y21 Y22 V2 where the individual components of the Y -matrix are found by:

I1 I1 Y11 = Y12 = V V 1 V2=0 2 V1=0 (3.60)

I2 I2 Y21 = Y22 = V1 V2=0 V2 V1=0

Although the impedance and admittance matrices describe the behaviour of a microwave network completely, it appears that this description is not very practical at microwave frequencies. This is due to the fact that we cannot measure the complex values of the total port voltages and currents in an accurate way at these frequencies. However, we can measure the amplitude and phase of the incident and reflected voltage waves in an accurate way by measuring the complex amplitude of the incident and reflected electric field of the TEM wave. This is done with a vector network analyser (VNA). Let us consider again the K-port microwave network, but now described in terms of the complex amplitude of the incident and reflected waves at each port. This is illustrated in Fig. 3.16. The normalized complex incident and reflected wave amplitudes are given by:

+ Vk +p ak = √ = Ik Z0, Z0 (3.61) − Vk −p bk = √ = Ik Z0. Z0

Note that Z0 = 50 Ω is usually used in measurements. The relation between the incident waves [a] and reflected waves [b] is determined by the scattering matrix [S], which for a two-port microwave 3.9. MICROWAVE NETWORKS 43

Microwave network

a1 b1 ak bk aK bK

Figure 3.16: K-port microwave network described using normalized complex amplitudes of the incident and reflected TEM waves network takes the following form:

     

 b1   S11 S12   a1          =     , (3.62)             b2 S21 S22 a2 where the individual scattering parameters are found by:

b1 b1 S11 = S12 = a a 1 a2=0 2 a1=0 (3.63)

b2 b2 S21 = A22 = a1 a2=0 a1 V1=0

So the scattering parameter S11 can be found by properly matching the output port 2. The other parameters are found in a similar way. In case of a one-port network, we find that the input reflection coefficient Γin = S11 Now consider the two-port microwave network of Fig. 3.17 with source impedance ZS at the input port and load impedance ZL at the output port. The parameter 2 2 |S21| is known as the forward gain and |S12| is the reverse gain. Note that in case of passive microwave networks both the forward and reverse gain are smaller than 1 and S21 = S12 due to the reciprocity principle. The input and output reflection coefficients of this network are found by:

b1 S12S21 Γin = = S11 + a1 1 − S22 ΓL (3.64) b2 S12S21 Γout = = S22 + , a2 1 − S11 ΓS 44 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

Γin 2 Γ Forward Gain (|S21| ) out

Z S a1 a2 Two-port ZL + Microwave Network b1 b2 VS -

Reverse Gain (|S |2) ΓS 12 ΓL

Figure 3.17: 2-port microwave network with input and output terminated with ZS and ZL, respec- tively

where ΓS and ΓL are the reflection coefficients when looking into the source and load, respectively:

ZS − Z0 ΓS = ZS + Z0 (3.65) ZL − Z0 ΓL = . ZL + Z0

Combining (3.64) and (3.65) we find that Γin = S11 only in the case that ZL = Z0. Similarly, we find that Γout = S22 when ZS = Z0. Exercise. Consider the two-port network of Fig. 3.18, where a 50 Ω resistor is connected in series between both ports. The source and load impedance are ZS = ZL = 50 Ω. Show that the scattering matrix takes the following form:     1 2  S11 S12       3 3    =   . (3.66)        2 1  S21 S22 3 3

3.10 Power Combiners

Power combiners and power splitters are one of the most commonly used microwave circuits. For example, in basestations for wireless communications, where a high effective isotropic radiated power (EIRP) is required to provide coverage of large macro-cells. A high EIRP can be achieved by combining the output power of several power amplifiers using a power combiner. Another application is in array antennas, where received signals from the individual antennas can be combined into a single output signal. An example of a 4-channel power combiner is shown in Fig. 3.19. Since power combiners are passive transmission line circuits, reciprocity holds. As a result, power combiners can Fig. 3.20 shows the basic power divider with unequal power division ratio α and a power combiner with equal power division ratio with α = 0.5. It can be shown 3.10. POWER COMBINERS 45

50 Ω

50 Ω a1 a2 50 Ω + b1 b2 VS -

Figure 3.18: 2-port microwave network consisting of a 50 Ω lumped element connected in series between input and output ports

Figure 3.19: Example of a power combiner/divider with an unequal power division ratio, a 4:1 Wilkinson combiner used in base-station antennas, courtesy Rob Mestrom

[12] that the three-port network of Fig. 3.20 cannot be at the same time lossless, reciprocal and matched at all ports at the same time. Therefore, any practical realization of a combiner or divider will be a compromise. In the rest of this section we will investigate several well known power combiner/divider types. Since the passive combiner/divider is a reciprocal device, we will only consider dividers in the remaining part of this section. Let us start by taking a closer look at the T-junction divider as illustrated in Fig. 3.21. An input transmission line with characteristic impedance Z0 is connected at the junction with two other transmission lines with characteristic impedance of Z1 and Z2, respectively. The additional susceptance jB represents the fringing fields and higher-order modes that might exist at the T- junction. For the sake of simplicity, we will assume that B = 0. The input admittance at the input of the T-junction is now simply:

1 1 (3.67) Yin = + . Z1 Z2 46 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

P2=αP1 P1 unequal Divider P3=(1-α) P1

P = 2 P1 P2 +P3 equal Combiner P3

Figure 3.20: Power divider with unequal power division ratio α (top) and a power combiner with equal power division ratio (bottom)

Ζ1

+ Ζ0 V0 jB -

Y in Ζ2

Figure 3.21: T-junction power divider with unequal power division ratio

1 For a matched input port it is required that Yin = . Now suppose that we would like to design Z0 1 a 2:1 divider with α = 3 and Z0 = 50 Ω. The input power delivered to the matched divider is:

2 V0 (3.68) Pin = . 2Z0

The corresponding output powers are given by:

2 V0 1 P1 = = αPin = Pin, 2Z1 3 (3.69) 2 V0 2 P2 = = (1 − α)Pin = Pin. 2Z2 3 3.10. POWER COMBINERS 47

Both equations can be satisfied with the following choice for the characteristic impedances:

Z1 = 3Z0 = 150 Ω, (3.70) 3Z Z = 0 = 75 Ω. 2 2

The input impedance is now indeed Zin = 50 Ω because of the parallel connection of a 75 Ω and 150 Ω impedance. However, the output ports are not matched and have a quite poor reflection coefficient:

Zout1 − Z1 30 − 150 Γ1 = = = −0.66, Zout1 + Z1 30 + 150 (3.71) Zout2 − Z2 37.5 − 75 Γ2 = = = −0.33. Zout2 + Z2 37.5 + 75 Note that poor output matching is not always a problem in an application. Another limitation of the simple T-junction divider is the poor isolation between the output ports, which could be a major issue in phased-array antennas. A way to improve the poor output matching and poor isolation is by adding a star connection of lumped-element resistors at the junction, as illustrated in Fig. 3.22. In this case, an equal-split divider is created. The impedance Z looking via the Z0/3 resistor into port 2 is given by:

port 2

P2 Ζ0/3 + Ζ0 Ζ0/3 port 1 V2 Ζ0/3 + - P1 Ζ0 V1 Ζ - + V3 Z - in Ζ0 P3 port 3

Figure 3.22: Resistive power divider with equal power division ratio α = 0.5

Z 4Z Z = 0 + Z = 0 . (3.72) 3 0 3

From the input port we then obtain a Z0/3 resistor in series with a parallel connection of Z at both output ports:

Z 1 4Z  Z = 0 + 0 = Z . (3.73) in 3 2 3 0 As a result, the input is well matched. Due to symmetry, the two output ports are also matched.

Therefore, all diagonal components of the scattering matrix [S] are zero: S11 = S22 = S33 = 0. 48 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

The other components of the scattering matrix can be found by determining the relation between the input voltage V1 and the output voltages V2 and V3. Since all ports are matched, the total voltage is equal to the voltage amplitude of the TEM-wave travelling in the +z direction, see also (3.61). The voltage at the central node is now easily found by using circuit theory:

2Z0/3 2V1 (3.74) V = V1 = . Z0/3 + 2Z0/3 3 The corresponding output votages at port 2 and 3 then become:

Z0 3V V1 (3.75) V2 = V3 = V = = . Z0 + Z0/3 4 2 As a result, the three-port scattering matrix takes the following form:      S11 S21 S31   0 1 1            1       [S] =  S S S  =  1 0 1  . (3.76)  21 22 23  2               S31 S32 S33 1 1 0

From (3.76) we observe that S21 = S31 = S23 = 1/2, which corresponds to a power level −6 dB 1 below the input power level. The output power at each of the output ports P = P = P . 2 3 4 in Therefore, 50% of the input power is dissipated in the resistors. This is a major drawback in most microwave applications. In addition, the isolation between the output ports is also quite poor. So far we have investigated power dividers with major limitations in terms of matching, isolation or power dissipation. A power divider which combines good matching, high isolation and low loss is the Wilkinson power divider. As we will see later, it is only lossless (assuming lossless transmission lines) for the incident power. The reflected power from the output ports is dissipated. Wilkinson dividers can be made for any division ratio, but we will only investigate the equal split case α = 0.5 in more detail in this section. Fig. 3.23 shows the transmission line circuit of the equal-split Wilkinson power divider and a realisation in microstrip technology. It consists of a T- √ junction with two quarter-wave transmission lines with Z1 = Z2 = 2Z0 which are connected via a lumped-element resistor R = 2Z0. The Wilkinson divider can be analysed by using the even-odd mode technique. In fact, we will excite the output ports with either the same voltage V (even mode) and with opposite voltage V and −V (odd mode). By using superposition, we then obtain the total voltage at the input port. From the voltages, we can determine the scattering matrix. The even-odd mode analysis starts by creating a new drawing that illustrates the symmetry of the Wilkinson power divider as shown in Fig. 3.24. All ports are terminated with a load impedance √ of Z0. Furthermore, we will assume that Z = 2Z0 and R = 2Z0. Lateron, we will show that with these particular values the best performance is obtained.

Even-mode analysis

In this case the source voltages at port 2 and port 3 are equal with Vs2 = Vs3 = 2Vs. Due to the symmetry in Fig. 3.24, no current will flow through the resistors R/2, which means that the 3.10. POWER COMBINERS 49

port 2 λ/4 Ζ0 Ζ1 port 1

R Ζ0

port 3

Ζ2 Ζ0 λ/4

√ Figure 3.23: Equal split Wilkinson power divider, Z1 = Z2 = 2Z0 and R = 2Z0

- V2 Z0 + Z port 2 + Vs2 2Z R/2 - 0 λ/4

port 1 Symmetry line + 2Z0 V1 R/2 - Z port 3 + + V Z0 3 Vs3 - -

Figure 3.24: Symmetrical representation of the equal-split Wilkinson power divider.

symmetry line can be represented as an open-circuit. As a result, we can limit our analysis to the circuit shown in Fig. 3.25. The resistor R/2 can now be neglected. The input impedance looking into port 2 is now easily found using the quarter-wave transformer equation (3.36):

√ 2 2 Z ( 2Z0) (3.77) Zin,even = = = Z0. 2Z0 2Z0 50 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

- V 2e Z0 port 2 + Z + - 2Vs - 2Z V1e R/2 0 + λ/4 port 1 open open circuit circuit Zin,even

z 0 −λ/4

Figure 3.25: Even-mode equivalent circuit of the equal-split Wilkinson power divider.

The voltage along the lossless λ/4 line with phase constant β is found using:   V (z) = V + e−βz + Γeβz . (3.78) As a result the port voltages are given by:

+ V2e = V (z = −λ/4) = V (1 − Γ) = Vs, (3.79) Γ + 1 V = V (0) = V + (1 + Γ) = V , 1e s Γ − 1 where Γ is the reflection coefficient looking at port 1 into the load impedance 2Z0: √ √ 2Z − 2Z 2 − 2 Γ = 0 √ 0 = √ . (3.80) 2Z0 + 2Z0 2 + 2 The voltage at port 1 now becomes

√ (3.81) V1e = −Vs 2.

Odd-mode analysis

The source voltages at port 2 and port 3 have now equal amplitude but opposite sign, Vs2 = 2Vs and Vs3 = −2Vs. Due to the asymmetry, the voltage along the symmetry line is zero as illustrated in the equivalent circuit of Fig. 3.26. The quarter-wave transformer now transforms the short at the end of the line to an open circuit at the start of the line at port 2. The remaining circuit is a voltage divider. Therefore, the voltages at port 1 and port 2 are given by:

V1o = 0, (3.82) 1 V = (2V ) = V . 2o 2 s s 3.11. IMPEDANCE MATCHING AND TUNING 51

- V 2o Z0 port 2 + Z + - 2Vs - 2Z V1o R/2 0 + λ/4 port 1

Figure 3.26: Odd-mode equivalent circuit of the equal-split Wilkinson power divider.

Total solution: superposition of even-odd modes By applying superposition of the even- and odd-mode analysis we obtain the total solution when exciting port 2. In a similar way, the solution when exciting port 3 can be found. In both the even and odd mode, port 2 is matched. Therefore S = S = 0. In addition, it can be shown √ 22 33 that with Z = 2Z0 port 1 is also matched S11 = 0. The transfer functions when all ports are matched are not found by superimposing the port voltages:

V1e + V1o − S12 = S21 = = √ , V2e + V2o 2 − (3.83) S13 = S31 = √ , 2

S23 = S32 = 0.

2 2 The ideal Wilkinson combiner has no losses since |S21| = |S12| = 1/2, so when used as a divider, 50% of the power is directed to each of the output ports. Furthermore, the isolation between the output ports is perfect since S23 = S32 = 0. In a practical realization of the Wilkinson divider, there will be losses due to metal and dielectric losses. In addition, the isolation (20 log10 |S32|) will be limited, typically to a value between -20 and -30 dB.

3.11 Impedance matching and tuning

Consider the circuit of Fig. 3.27 where a source with complex source impedance ZS = RS + XS is connected to a complex load ZL = RL + XL. The source could be a power amplifier and the load could be an antenna. The time-average power delivered to the load is found by:

1 P = Re {V I∗ } . (3.84) L 2 L L 52 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS where

VSZL VL = , ZS + ZL (3.85) VS IL = . ZS + ZL

IL ZS + +

VS ZL VL - -

Figure 3.27: Conjugate matching of a source impedance to a complex load impedance.

When substituting (3.85) in (3.84) we obtain:

( 2 ) 2 1 |VS| ZL 1 |VS| RL (3.86) PL = Re 2 = 2 2 . 2 |ZS + ZL| 2 (RS + RL) + (XS + XL)

The maximum delivered power is now found by determining the value of (RL,XL) for which the partial derivatives are equal to zero:

∂P ∂P L = L = 0. (3.87) ∂RL ∂XL

∗ We find that this occurs when RL = RS and XL = −XS, in other words when ZL = ZS. The maximum power transfer to the load occurs when the source and load impedances are conjugate matched to each other. In a lot of situations it will appear that direct conjugate matching of the source to load is not possible. In these cases, we can use a matching circuit as illustrated in Fig. 3.28 The matching circuit can be created using either lumped elements like inductors (with inductance L [H])and capacitors (with capacitance C [F]) or by using distributed transmission- line components. Note that lumped elements can only be used when the size of the component is much smaller as compared to the wavelength. In PCB technologies, surface-mount devices (SMD) are commonly used, with sizes down to SMD0201 (0.25 mm × 0.125 mm) and even SMD01005 (0.125 mm × 0.0675 mm). SMD components are typically used at frequencies up to 6 GHz. In semiconductor technologies, much smaller inductors and capacitors can be realized. As a result, lumped-element matching can be used up to much higher frequencies in integrated circuits (ICs). Lumped-element matching can be realized by placing inductors and capacitors in series or parallel 3.11. IMPEDANCE MATCHING AND TUNING 53

ZS

+ Matching V Z S circuit L -

Figure 3.28: Matching circuit to obtain a good matching between the source and load. to the load impedance. The impedance of both elements in the frequency domain are given by:

1 capacitor : Z = , ωC (3.88) inductor : Z = ωL, where ω = 2πf. The Smith chart is a useful tool to design such a network. Several open-source tools are available to design lumped element circuits, e.g. http://www.fritz.dellsperger.net/smith.html.

Suppose that we want to match a particular load impedance ZL to a source impedance ZS. Fig. 3.29 shows the degrees of freedom when placing an inductor or capacitor in series or in shunt (parallel) to the load impedance. By using several of these combinations (shunt-series) or (series- shunt), we can reach any point in the smith chart. Note that ideal capacitors and inductors do not exist in practise at microwave frequencies. One has to take parasitics and losses into account. This will deteriorate the performance of a lumped-element matching circuit.

Exercise. Design a lumped element matching circuit to match a complex load impedance ZL = 200 − 100 Ω to a real load ZS = 100 Ω. The frequency of operation is 1 GHz. You can use the design tools ADS or QUCS (open-source, http://qucs.sourceforge.net/). Check you final result in a Smith chart using the strategy as shown in Fig. 3.29.

Due to the practical limitation of lumped-elements, at microwave frequencies we generally prefer to use transmission-based matching circuits. When both the load and source impedance are real, we can use the quarter-wave transformer as discussed in section 3.5. Another matching strategy is to use stubs which are placed in parallel to a transmission line transformer. An example of such a circuit is shown in Fig. 3.30. The stub is either open ended or shorted. The imput impedance is then obtained using (3.27):

shorted stub : Zs = Z0 tan βl = ΩZ0 = ΩL, (3.89) Z Z 1 open stub : Z = 0 = 0 = . s  tan βl Ω ΩC Apparently, a shorted stub is the equivalent of an inductor and an open stub is the equivalent of 54 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

C

Series C ZL Shunt L Series L ZL

Series C Shunt C Shunt C C ZL

L

Series L ZL

Shunt L L ZL

Figure 3.29: Lumped-element matching illustrated in a Smith chart.

d

Z0 Z0 ZL

Z0

l YT=1/ZT open or shorted stub

Figure 3.30: Single-stub matching circuit

a capacitor, with l ≤ λ/4. This equivalence appears to be very useful when designing microwave filters, as we will discuss in more detail in section 3.12.

Impedance matching with the single-stub tuner of Fig. 3.30 is now achieved as follows. Let

YT = 1/ZT be the admittance of the terminated transmission line with length d. We can determine 3.12. MICROWAVE FILTERS 55

YT using equation (3.27):

ZL + Z0 tan βd ZT = Z0 , Z0 + ZL tan βd (3.90) 1 YT = = GT + BT . ZT

Now choose the length d in such a way that GT = Y0. Furthermore, the shunt stub is designed to cancel the imaginary part of YT , so Ys = −BT . Next to single-stub tuners, double stub tuners can be used to increase the degrees of freedom. One of the additional advantages of a double-stub tuner is that it can be designed in such a way that the line length between the two stubs can be fixed, which simplifies the construction of such a tuner.

Exercise. Determine the parameters (l, d) of a single-stub tuner with a shorted stub to match a complex load impedance ZL = 60 − 80 to a 50 Ω transmission line. Use transmission lines with Z0 = 50 Ω. Frequency of operation is 1 GHz. Answer: d = 33mm and l = 28.5mm.

3.12 Microwave filters

Microwave filters can be used to manipulate the frequency response of a microwave system. Mi- crowave filters are constructed using transmission line components. At lower frequencies (< 5 GHz) lumped elements (L, C) can be used as well. Well-known filter characteristics are low-pass, high-pass, bandpass or bandstop. Fig. 3.31 illustrates these 4 types using lumped elements. There is a lot of literature available for designing lumped-element filters [12]. In this section, we will assume that such an ideal lumped-element prototype filter is already known. We will show how such a lumped-element filter can be transformed into a microwave filter using transmission line components. In the previous section in (3.89) we have already shown that shorted or open stubs

Low-pass High-pass Band-pass Band-stop

Figure 3.31: Lumped-element filters. are equivalent to inductors and capacitors. With the particular choice that the length of the stub 56 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

2π λ π l = λ/8 we find that Ω = tan βl = tan = tan = 1 which results in: λ 8 4

shorted stub : Zs = Z0 = L, (3.91) Z 1 open stub : Z = 0 = . s  C Which resembles the equivalence of an inductor with inductance L and capacitor with capacitance C at ω = 1, respectively. Figure 3.32 illustrates the equivalence between the open/shorted stub and L,C lumped elements for any value of the angular frequency ω = 2πf. Stubs can be used in

l=λ/8 at ω

Short Z =ωL L 0 Circuit Z =jωL Zin=jωL in

l=λ/8 at ω

Open C Y =ωC 0 Circuit Yin=jωC Yin=jωC

Figure 3.32: Equivalence of lumped-element L,C and shorted/open stubs with length of λ/8. This is also known as Richard’s transformation. parallel (see Fig. 3.30) or in series to a transmission line. A parallel stub can be easily realized in a practical PCB technology, for example as a shorted or open microstrip line. A series stub cannot be easily realized. Therefore, it is most common to transform series stubs to parallel stubs using the so-called Kuroda’s identities. Fig. 3.33 shows the four identities. We will now show how Kuroda’s identity can be used to design a low-pass filter in microstrip technology. We will start with a lumped-element prototype, corresponding to a third-order maximally flat low-pass filter prototype (see [12], Table 8.3). The ideal circuit model and the corresponding steps towards the final transmission-line equivalent is provided in Fig. 3.34. The filter has a -3 dB roll-off at 10 GHz. We have assumed that both the input and output of the circuits are connected to 50Ω ports. The design starts with the ideal circuit model using two inductors and a parallel capacitor. Next the lumped elements are transformed to transmission line sections, in line with Fig. 3.32. Finally, by adding additional λ/8 line sections, we can apply Kuroda’s identity (see Fig. 3.33). The final design can be easily realized in microstrip technology on a PCB. The corresponding simulated performance of the original ideal circuit model and the final transmission-line low-pass filter is shown in Fig. 3.35. 3.12. MICROWAVE FILTERS 57

2 Z1/n l l

2 1/Z2 Z1 Z2/n

Z 1 l l

2 2 Z2 n Z1 1/(n Z2)

l l 1:n2

Z 2 2 1 Z2 Z2/n Z1/n

2 1/Z2 1/(n Z2) l l n2:1

2 Z1 n Z1

Z Figure 3.33: Kuroda’s identities. n2 = 1 + 2 , l = λ/8. Z1

λ/8 λ/8 Z0=50.3 Ω Z0=50.3 Ω L1=0.8 nH L2=0.8 nH

λ/8 λ/8

C1=0.64 pF

Z =50 Ω Z0=50 Ω 0

Z0=24.9 Ω λ/8

Z0=50.3 Ω λ/8 λ/8 Z0=50.3 Ω

Z0=100.3 Ω Z0=100.3 Ω

λ/8 λ/8

Z0=24.9 Ω Ω λ λ λ λ/8 Z0=99.7 /8 Z0=24.9 Ω /8 Z0=99.7 Ω /8

Figure 3.34: Design of a transmission-line low-pass filter in 4 steps using Richard’s transformation 2 and Kuroda’s identities with n = 1.995. Maximally-flat filter with N = 3, R0 = 50Ω and f−3dB = 10 GHz 58 CHAPTER 3. TRANSMISSION LINE THEORY AND MICROWAVE CIRCUITS

Figure 3.35: Simulated performance of the low-pass filter, ideal lumped element circuit (red) and transmission-line equivalent (blue). Chapter 4

Antenna Theory

In this chapter we will introduce the general properties of radiated fields from antenna sources. We will use Maxwell’s equations in the space-frequency domain as a starting point in which the antenna is represented by a current or charge distribution. Next, we will determine the electromagnetic field in the far field of an arbitrarily electric current or charge distribution. Based on this, several elementary antenna structures will be analyzed, such as wire antennas and loop antennas. Finally, we will introduce magnetic sources and the Lorentz-Lamor theorem in order to extend the theoretical framework towards the analysis of aperture antennas, such as horns, reflector antennas and microstrip antennas.

4.1 Maxwell’s equations

Maxwell’s equations describe the propagation of electromagnetic waves generated by a current or charge distribution in any medium. In the time domain Maxwell’s equations are given by

∂B~(~r, t) ∇ × E~(~r, t) = − , ∂t ∂D~ (~r, t) ∇ × H~ (~r, t) = + J~e(~r, t), ∂t (4.1) ∇ · B~(~r, t) = 0,

∇ · D~ (~r, t) = ρe(~r, t), where E~ is the electric field in Volt per meter [V/m], D~ the displacement current in Coulomb per m2 [C/m2], B~ the magnetic flux density in Weber per m2 [W/m2], H~ the magnetic field in 3 3 Ampere per meter [A/m], ρe the electric charge distribution in Coulomb per m [C/m ] and J~e the electric current distribution in Ampere per m2 [A/m2]. In addition, the continuity equation provides a relation between the electric current and charge distribution:

∂ρ (~r, t) ∇ · J~ (~r, t) + e = 0. (4.2) e ∂t

59 60 CHAPTER 4. ANTENNA THEORY

Note that the continuity equation is not an independent equation, but can be derived from Maxwell’s equations (4.1). Since the propagation of electromagnetic waves from a transmit antenna to a receive antenna usually takes place in free-space, we can state that:

D~ (~r, t) = 0E~(~r, t), (4.3)

B~(~r, t) = µ0H~ (~r, t).

Furthermore, we will again assume a time-harmonic dependence of all field components, currents and charges, so that they depend on the angular frequency ω = 2πf, where f is the frequency expressed in Hertz [Hz]. In this way, the electric field is expressed as:

h i E~(~r, t) = Re E~ (~r)eωt . (4.4)

A similar relation holds for the other components of (4.1). Maxwell’s equations now reduce to the following form:

∇ × E~ (~r) = −ωµ0H~ (~r),

∇ × H~ (~r) = ω0E~ (~r) + J~e(~r), (4.5) ∇ · H~ (~r) = 0,

ρ (~r) ∇ · E~ (~r) = e . 0 Next step is now to determine the electric and magnetic fields, E~ (~r) and H~ (~r), due to the source distributions, given by J~e(~r) and ρe(~r). From these frequency-domain fields, we can then deter- mine the physical (time-domain) fields E~(~r, t) and H~ (~r, t) by using relation (4.4). Note that in 4.5, we have used the linear properties of Maxwell’s equations, which allows to interchange the vector operators ∇× and ∇· with the operator Re[]. In the rest of the derivation, we will omit the use of (~r) in our notation. From ∇ · H~ = 0 we can conclude that H~ can be expressed in terms of a vector potential A~e according to 1 H~ = ∇ × A~e. (4.6) µ0 ~ ~1 ~ Note that Ae does not have a unique solution, since the vector Ae = Ae + ∇φe, with φe an arbitrarily scalar potential, provides exactly the same solution of H~ . Substitution of (4.6) in the first equation of (4.5) provides

∇ × E~ = −ω∇ × A~e, (4.7) which leads to

E~ = −ωA~e − ∇φe, (4.8) 4.1. MAXWELL’S EQUATIONS 61 where φe is an unknown scalar function of ~r. Substituting (4.8) in the second equation of (4.5) gives

~ ~ ~ 2 ~ ~ ∇ × ∇ × Ae = ω0µ0E + µ0Je = k0Ae − ω0µ0∇φe + µ0Je. (4.9)

2 2 2 2 Where k0 is the wavenumber in free space with k0 = ω 0µ0 = ω /c , with c the speed of light. 2 By using the vector property ∇ × ∇ × A~e = ∇∇ · A~e − ∇ A~e, (4.9) becomes 2 ~ 2 ~ ~ ~ ∇ Ae + k0Ae = ∇∇ · Ae + ω0µ0∇φe − µ0Je. (4.10)

Up to now, we did not specify the scalar potential φe. We will choose φe according to

∇ · A~e = −ω0µ0φe. (4.11)

This particular choice is known as the Lorentz-gauge. Equation (4.10) can now be re-written in the well-known Helmholtz equation

2 ~ 2 ~ ~ ∇ Ae + k0Ae = −µ0Je. (4.12) Note that we also need to satisfy the last equation of (4.5). This results in

2 ∇ · (0E~ ) = −ω0∇ · A~e − 0∇ φe = ρe. (4.13)

Again, we apply the Lorentz-gauge, which provides us

2 2 ρe ∇ φe + k0φe = − . (4.14) 0

When considering antennas, the sources which are represented by the electric current density J~e and charge distribution ρe, will be located within a finite-size volume V0. This is illustrated in Fig. 4.1. As a first step, we will determine the solution of the Helmholtz equation for an electric dipole (point source) oriented along the z-axis located at the origin of the coordinate system (x, y, z) = (0, 0, 0)

, where ~ez represents the unit vector along the z-direction. The configuration is illustrated in Fig. 4.2. Now let I0 be the total current flowing through the dipole and l the length of the dipole. Characteristic for the electric dipole is the fact that the length is much smaller than the free-space wavelength, so l  λ0. The product I0l = p is also called the ”dipole-moment”. The current density along the electric dipole can now te expressed as:

J~e = I0lδ(x)δ(y)δ(z)~ez = I0lδ(~r)~ez. (4.15)

Here δ(r) represents the three-dimensional delta function. Since the current only has a component in the z direction, the vector potential will also only have a component in the z-direction, i.e.

A~e = Aez~ez. The vector potential now satisfies the following equation

2 2 ∇ Aez + k0Aez = −µ0I0lδ(~r). (4.16)

The scalar quantity Aez needs to represent a spherical behavior, since we have assumed a point source. As a result Aez(~r) = Aez(r). Therefore, we can rewrite (4.16) in the following form 1 d  dA  r2 ez + k2A~ = −µ I lδ(~r). (4.17) r2 dr dr 0 ez 0 0 62 CHAPTER 4. ANTENNA THEORY

z

P

rd Q r

ro

O y

V

x

Figure 4.1: The antenna current and charge distribution are located within the volume V0

z

P

r

J l e

y

x

Figure 4.2: Electric dipole located in the origin and oriented along the z-axis with J~e = I0l~ez.

Outside the origin, we now find two possible solutions of this differential equation: C 1 k0r Aez = e , r (4.18) C A2 = e−k0r. ez r 4.1. MAXWELL’S EQUATIONS 63

1 where C is a constant. The first solution Aez represents a spherical wave propagating towards the 2 source, whereas the second solution Aez represent a wave which propagates away from the source. This is due to the fact that we have assumed a eωt the time dependence of the fields. We will use the second solution, where we still need to determine the constant C. We can determine C by substituting k0 ↓ 0 in (4.17), which transform in to the Poisson equation known from electrostatics with solution

µ I l A = 0 0 . (4.19) ez 4πr

µ I l As a result, we find that C = 0 0 . The general solution of the vector potential A~ of an electric 4π e dipole along the z-axis now takes the following form:

µ I l A~ = ~e A = ~e 0 0 e−k0r. (4.20) e z ez z 4πr The vector potential in (4.20) has the same direction as the dipole. This can be generalized to all directions. Therefore, the vector potential of an arbitrarily oriented dipole located at

~r0 = ~exx0 + ~eyy0 + ~ezz0 can be found by substituting r in expression (4.20) by |~r − ~r0| = p 2 2 2 (x − x0) + (y − y0) + (z − z0) . We now obtain

−k|~r−~r | µ0 e 0 A~e(~r) = I0~l(~r0) 4π |~r − ~r0| (4.21)

= µ0I0~l(~r0)G(~r, ~r0).

The function G(~r, ~r0) is known as the Greens function and represents the response of a point source. The Greens function plays an important role in numerical electromagnetic methods (e.g. Method-of-Moments) which are used to analyse complex antenna structures. More details can be found in chapter 5.

The vector potential of a general electric current distribution J~e within the volume V0 can be obtained by applying the superposition principle. The solutions of expression (4.12) and (4.14) for an observation point P outside the source volume V0 can now be easily expressed by:

−k |~r−~r | µ0 Z e 0 0 A~e(~r) = J~e(~r0) dV0, (4.22) 4π V0 |~r − ~r0|

1 Z e−k0|~r−~r0| φe(~r) = ρe(~r0) dV0, (4.23) 4π0 V0 |~r − ~r0| where ~r0 indicates a source coordinate in the volume V0 and ~r indicates an observation point at which the fields are being determined. It should be noted that the solutions of (4.12) and (4.14) should meet the Lorentz-gauge. By using the notation

1 e−k0|~r−~r0| ϕ(~r, ~r0) = , (4.24) 4π |~r − ~r0| 64 CHAPTER 4. ANTENNA THEORY we find that

∇r · J~e(~r0)ϕ(~r, ~r0) = ϕ(~r, ~r0)∇r · J~e(~r0) + (J~e(~r0) · ∇rϕ(~r, ~r0))

(4.25) = J~e(~r0) · ∇rϕ(~r, ~r0)

~ = −Je(~r0) · ∇r0 ϕ(~r, ~r0).

Note that the operator ∇r is related to ~r and not to ~r0. Furthermore, we have used the fact that ~ ∇rϕ(~r, ~r0) = −∇r0 ϕ(~r, ~r0). As a next step, we will determine ∇r · Ae.

Z ∇r · A~e(~r) = µ0 ∇r · J~e(~r0)ϕ(~r, ~r0)dV0 V0 (4.26) Z Z ~ ~ = µ0 ϕ(~r, ~r0)∇r0 · Je(~r0)dV0 − µ0 ∇r0 · Je(~r0)ϕ(~r, ~r0)dV0 V0 V0

By applying Gauss theorem and by selecting a volume V0 much larger than the source region, the second integral on the right-hand side of equation (4.26) will vanish. In the first integral we can replace J~e(~r0) by means of the continuity equation (4.2) by ρe. This results in the Lorentz-gauge:

Z ∇r · A~e(~r) = −ωµ0 ~ρe(~r0)ϕ(~r, ~r0)dV0 V0 (4.27)

= −ω0µ0φe(~r).

By using the Lorentz-gauge we are able to express the fields E~ and H~ directly in terms of the vector potential A~e. 1 H~ = ∇ × A~e, (4.28) µ0

E~ = −ωA~e − ∇φe

∇∇ · A~e = −ωA~e + ω0µ0 (4.29) 1 h 2 2 i = ∇ × ∇ × A~e + ∇ A~e + k A~e ω0µ0 1 h i = ∇ × ∇ × A~e − µ0J~e . ω0µ0 Outside the source region, we now find that

1 H~ = ∇ × A~e, µ0 (4.30) 1 E~ = ∇ × ∇ × A~e. ω0µ0 4.2. RADIATED FIELDS AND FAR-FIELD APPROXIMATION 65

Since the vector potential A~e only depends on J~e (and does not depend on the charge distribution ρe), we only need to know the electric current distribution J~e on a particular antenna structure in order to determine the radiated electromagnetic fields. This is a consequence of the applied Lorentz-gauge.

4.2 Radiated fields and far-field approximation

In chapter 2 we have discussed the various field regions of antennas. Since most antennas are used to transfer information over long distances, we are primarily interested in the fields far away from the antenna. In this section we will determine a general analytic expression of the electromagnetic field in the so-called far-field region of the antenna. The derivation starts with the expression of the vector potential (4.22), which we derived in the previous section. Combining this with (4.30) we obtain

1 H~ = ∇ × A~e µ0 (4.31) Z = ∇r × J~e(~r0)ϕ(~r, ~r0)dV0, V0 where ϕ has been defined in 4.24. Furthermore, operator ∇r× is performed on the field point ~r. We now know that

∇r × J~e(~r0)ϕ(~r, ~r0) = ϕ(~r, ~r0)∇r × J~e(~r0) + ∇rϕ(~r, ~r0) × J~e(~r0). (4.32)

Note that ∇r × J~e(~r0) = 0 and 1  1  1   −k0|~r−~r0| −k0|~r−~r0| ∇rϕ(~r, ~r0) = e ∇r + ∇r e . (4.33) 4π |~r − ~r0| 4π|~r − ~r0| By using the following relations   1 ~r − ~r0 ∇r = − 3 , (4.34) |~r − ~r0| |~r − ~r0| and   (~r − ~r ) −k0|~r−~r0| −k0|~r−~r0| 0 ∇r e = −k0e , (4.35) |~r − ~r0| we find that

Z   −k0rd ~ 1 1 e ~ (4.36) H = −k0 − ~erd × Je(~r0)dV0, 4π V0 rd rd where we have introduced the coordinate ~rd according to ~rd = ~r − ~r0 with rd = |~rd| and where ~rd the unit-vector in the direction ~rd is given by ~erd = . rd We will now derive an approximation of the magnetic field H~ , which is only valid at a large distance from the antenna system. The distance rd between a field point P with coordinates (x, y, z) and the source point with coordinates (x0, y0, z0) is given by q 2 2 2 rd = ((x − x0) + (y − y0) + (z − z0) ). (4.37) 66 CHAPTER 4. ANTENNA THEORY

This can also be written in the following form q 2 2 2 2
rd = r − 2(xx0 + yy0 + zz0) + (x0 + y0 + z0) , (4.38)

2 2 2 2 with r = x + y + z . In the far-field region x0, y0 and z0 will be very small as compared to r. As a result, we can express rd in terms of a binomial distribution according to " ! 1 2 x2 + y2 + z2 r = r 1 + − (xx + yy + zz ) + 0 0 0 d 2 r2 0 0 0 r2  2 2 2 !2 1 2 x0 + y0 + z0 − − (xx0 + yy0 + zz0) + + · · ·· 8 r2 r2 " # xx + yy + zz x2 + y2 + z2 (xx + yy + zz )2 (4.39) = r 1 − 0 0 0 + 0 0 0 − 0 0 0 + ··· r2 2r2 2r4 xx + yy + zz ≈ r − 0 0 0 r

= r − (~er, ~r0),

~r x2 + y2 + z2 with ~e = . This approximation is only valid if the term 0 0 0 is small enough. The r r 2r corresponding exponential factor will be close to 1. The criterion which is often used in literature is x2 + y2 + z2 λ 0 0 0 < 0 . (4.40) 2r 16 Consider an antenna with maximum dimension L and assume that the center of the antenna is located in the origin of our coordinate system. Equation(4.40)now corresponds to the well-known far-field criterion introduced in chapter 2 2L2 R > . (4.41) λ0

e−k0rd We can now also replace by r in expression (4.36) when the following conditions are rd fulfilled

2 max(xx0 + yy0 + zz0)  r , (4.42) and

2 2 2 2 max(x0 + y0 + z0)  r . (4.43) The second condition is fulfilled when r  L. In addition, the first condition is then also fulfilled  1  since x/r, y/r and z/r are both of order 1. If r  λ0, we can replace −k0 + by −k0. By rd applying these approximations, we now obtain the following expression for the magnetic field H~

−k e−k0r Z ~ 0 ~ k0(~er,~r0) (4.44) H = ~erd × Je(~r0)e dV0, 4πr V0 4.2. RADIATED FIELDS AND FAR-FIELD APPROXIMATION 67

If our observation point P is located far away from the antenna, the vector ~erd and vector ~er will be parallel. As a consequence, we can replace ~erd by ~er. The unit vector ~er does not depend on the source point Q and can, therefore, by moved outside the integral. We then finally obtain

−k e−k0r Z 0 k0(~er,~r0) (4.45) H~ = ~er × J~e(~r0)e dV0, 4πr V0 Expression (4.45) leads to several important conclusions:

~ 1 i. H has an r dependency.

ii. From e−k0r and the time-harmonic behavior eωt, we can conclude that the waves propagate away from the source. iii. H~ · ~er = 0. This implies that the radial component vanishes in the far-field region. iv. The magnetic field can now be expressed as

e−k0r H~ = (~e H (θ, φ) + ~e H (θ, φ)). (4.46) r θ θ φ φ

The components Hθ(θ, φ) en Hφ(θ, φ)) can be determined in the following way. We can express the electric current density in the following form

J~e = Jer~er + Jeθ~eθ + Jeφ~eφ. (4.47)

We then obtain

~er × J~e = Jeθ~eφ − Jeφ~eθ. (4.48)

The field H~ can now also be written as

−k e−k0r Z 0 k0(~er,~r0) H~ = (J~eθ~eφ − Jeφ~eθ)e dV0, (4.49) 4πr V0

This specific formulation of the far field was already used in (2.5) in order to determine the radiation pattern of an antenna.

v. The field H~ in the observation point P can be interpreted as the superposition of the contri- butions from all source points. Note that the term ek0(~er,~r0) corresponds to a phase difference

between source point Q (x0, y0, z0) and the origin (0, 0, 0)

The final step in the derivation of the radiated fields from a general antenna is the determination of the electric field E~ (~r) using expresssion (4.30)

1 (4.50) E~ = ∇ × ∇ × A~e. ω0µ0 68 CHAPTER 4. ANTENNA THEORY

The vector operator ∇ × A~e was already applied previously resulting in (4.36). The electric field E~ now takes the form " # Z   −k0rd ~ 1 1 e ~ (4.51) E = ∇r × −k0 − 2 ~rd × Je(~r0) dV0, 4πω0 V0 rd rd

Note that the operator ∇r is not only a function of the vector ~r0 but also of ~r due to the factor   −k0rd 1 e ~ ~rd = ~r − ~r0. With ψ(~r, ~r0) = −k0 − 2 en ~a(~r, ~r0) = ~rd × Je(~r0), we obtain rd rd

∇r × ψ~a = ∇rψ × ~a + ψ∇r × ~a,

h i ∇r × ~a = ∇r × (~r − ~r0) × J~e(~r0)

= −2J~e(~r0), (4.52) !  1  e−k0rd e−k0rd  1  ∇rψ = −k0 − ∇r 2 + 2 ∇r −k0 − rd rd rd rd !  1  e−k0rd e−k0rd 1 = −k0 − ∇r 2 + 2 2 ~erd . rd rd rd rd Furthermore, ! e−k0rd 1 1 −k0rd −k0rd (4.53) ∇r 2 = −2e 3 ~erd + 2 (−k0)e ~erd . rd rd rd By substituting this expression in (4.52) we obtain

−k0rd   e 2 3 3 (4.54) ∇rψ = 2 k0 −1 − + 2 ~erd . rd krd (k0rd) When we apply the same kind of far-field approximation as used for H~ we find that

−k2 e−k0r Z h i ~ 0 ~ k0(~er,~r0) (4.55) E = ~erd × ~erd × Je(~r0) e dV0. ω0 4πr V0 ~ By substituting ~erd by ~er, we finally arrive at the expression for E in the far-field region of the antenna

−k2 e−k0r Z 0 k0(~er,~r0) (4.56) E~ = ~er × ~er × J~e(~r0)e dV0. ω0 4πr V0 Note that the expressions of E~ and H~ given by (4.56) en (4.45) can also be obtained by substituting the operator ∇× in equation (4.30) by (−k0)~er×. Applying the vector identity A~ × [B~ × C~ ] = (A~ · C~ )B~ − (A~ · B~ )C~ to expression (4.56) we find

−k2 e−k0r Z h i E~ = 0 (~e · J~ (~r ))~e − J~ (~r ) ek0(~er,~r0)dV ω 4πr r e 0 r e 0 0 0 V0 (4.57) k2 e−k0r Z 0 k0(~er,~r0) = (~eθJeθ(~r0)) + ~eφJeφ(~r0)) e dV0. ω0 4πr V0 4.3. ELECTRIC DIPOLE 69

From (4.57) we can conclude that E~ · ~er = 0, which implies that the radial component of E~ vanishes in the far-field region of the antenna. As a result the electric field E~ is perpendicular to

~er. By combining expression (4.45) and (4.56) we can observe that

E~ = Z0H~ × ~er, (4.58) 1 H~ = ~er × E,~ Z0 q where Z = µ0 ≈ 377Ω is the free-space wave impedance. From (4.58) we find that the 0 0 electromagnetic waves in the far-field region behave locally as TEM waves. The Pointing vector is now readily found from (4.58)

1 h i S~(~r) = Re E~ × H~ ∗ 2

1 h ∗ i = Re E~ × (~er × E~ ) 2Z0 (4.59) 1 h ∗ ∗i = Re (E~ · E~ )~er − (E~ · ~er)E~ 2Z0

1 2 = |E~ | ~er, 2Z0 sincet E~ · ~er. As expected, the electromagnetic energy indeed flows in the radial direction.

Summary For an antenna described by an electric current distribution J~e(~r0) within a finite volume V0, we can express the corresponding electromagnetic fields in the far-field region by

−k e−k0r Z 0 k0(~er,~r0) H~ = ~er × J~e(~r0)e dV0, 4πr V0 −k2 e−k0r Z E~ = 0 ~e × ~e × J~ (~r )ek0(~er,~r0)dV , ω 4πr r r e 0 0 0 V0 (4.60)

E~ = Z0H~ × ~er,

rµ0 Z0 = ≈ 377Ω. 0

4.3 Electric dipole

The most elementary antenna is the electric dipole. The electric dipole is an infinite small current element, which can be approximated in practice by a small linear antenna with a length much smaller than the free-space wavelength λ0. The electric dipole is located in the origin and directed 70 CHAPTER 4. ANTENNA THEORY along the z-axis, as indicated in Fig. 4.2. The current distribution J~e along the electric dipole can now be expressed in terms of the delta function:

J~e = I0lδ(x)δ(y)δ(z)~ez = I0lδ(~r)~ez. (4.61)

Where I0 is the total current which we assume to be constant along the electric dipole. The length of the antenna is l. The delta-function δ(z) belongs to the class of generalized functions with the property: ∞ Z γ(z0) = δ(z − z0)γ(z)dz. (4.62) −∞ The delta-function can be visualized as follows   1  w w  − 2 < z < 2 δ(z) = w , (4.63)   w  0 |z| > 2 where w needs to be very small. In order to determine whether (4.61) indeed represent an electric dipole, we need to investigate the corresponding charge distribution ρe which can be found by applying the continuity equation (4.2): 1 −I l ρ (~r) = − ∇ · J~ (~r) = 0 δ(x)δ(y)δ0(z). (4.64) e ω e ω By considering the derivative of (4.63), δ0(z), we can represent(4.64) by a set of two point charges with opposite sign and located on the z-axis at z = w/2 and −w/2, respectively, since 1   w   w  δ0(z) = δ z + − δ z − . (4.65) w 2 2 We can now determine the electromagnetic fields in the far-field region of the electric dipole by using the general expressions derived in chapter 4.2:

−k e−k0r Z H~ = 0 ~e × J~ (~r )ek0(~er,~r0)dV , 4πr r e 0 0 V0 (4.66)

E~ = Z0H~ × ~er,

Substituting the current distribution of the electric dipole (4.61) in the above equation and using the fact that

~er × ~ez = −~eφ sin θ (4.67) leads us to the magnetic field far away from the dipole

k I le−k0r H~ = H ~e = 0 0 sin θ~e . (4.68) φ φ 4πr φ The corresponding electric field in the far-field region is now given by

k Z I le−k0r E~ = E ~e = 0 0 0 sin θ~e , (4.69) θ θ 4πr θ 4.3. ELECTRIC DIPOLE 71 where we have used the following relation

~eφ × ~er = ~eθ. (4.70) The time-average power flux density (Pointing vector) expressed in [W/m2] is now found by using equation (4.59), which provides

1 2 S~(~r) = |E~ | ~er 2Z0 (4.71) 1 k2Z (I l)2 = 0 0 0 sin2 θ~e . 2 (4πr)2 r From (4.71) we can conclude that the electric dipole does not radiate uniformly in all directions. In the direction along the z-axis (θ = 0) the dipole does not radiate al all. The power density S~(~r) decreases as r−2 with increasing distance from the dipole. However, since the total power radiates through a sphere with radius r and surface 4πr2, the total radiated power is independent of r. The total radiated power by the electric dipole is then found as,

ZZ 2 Pt = S~(~r) · ~err dΩ 2π π Z Z 1 k2Z (I l)2 = 0 0 0 sin3 θdθdφ (4.72) 2 (4π)2 0 0 k2Z (I l)2 = 0 0 0 , 12π where we have used the fact that π Z 4 sin3 θdθ = . (4.73) 3 0

As expected, the total power Pt does not depend on r. Let us now explore some fundamental antenna parameters of the electric dipole According to (2.10), the normalized radiation pattern is given by P (θ, φ) P (θ) F (θ, φ) = = = sin2 θ. (4.74) P (π/2, 0) P (π/2) This pattern is plotted in Fig. 4.3. The 3-dB beam width of the electric dipole is 900. The directivity is the maximum of the directivity function, given in (2.11). We then obtain P (θ, φ) 3 D(θ, φ) = = sin2 θ, (4.75) Pt/4π 2 3 which implies that the directivity is D = . 2 Finally, let us determine the input impedance of the electric dipole. The input impedance of the electric dipole has only a real value Ra, also known as the radiation resistance, which can be found by using relation (2.16). Combining this with (4.72) we arrive at P k2Z l2  l 2 R = t = 2 0 0 = 80π2 . (4.76) a 1 2 2 |I0| 12π λ0 72 CHAPTER 4. ANTENNA THEORY

90 1 120 60

0.8

0.6 150 30 0.4

0.2

180 0

210 330

240 300

270

Figure 4.3: Radiation pattern versus θ of an electric dipole of length l. Note that θ = 00 coincides with the positive z-axis

The length l of the dipole is small as compared to the wavelength. For example, let l = 0.01λ0, which results in an input impedance of Ra = 0.08Ω. This imples that the electric dipole has a very low radiation resistance which will makes it difficult to match to a transmission line since most t commonly used transmission lines have a much higher characteristic impedance (often Z0 = 50Ω). This will result in a poorly matched antenna and a high corresponding reflection coefficient (2.19) at the antenna port. Electric dipoles are only used in very specific applications in which power- matching is not important. A nice example is low-frequency radio-astronomy [13]. The half-wavelength dipole has much better radiation and matching properties. This type of antenna will be explored in the next section.

4.4 Thin-wire antennas

In the previous section we found that the radiation resistance of an electric dipole is very small, which makes it difficult to match to a transmission line. The matching properties can be improved by increasing the length of the dipole, resulting in a so-called linear wire antenna. This antenna consists of a cylindrical metal wire with diameter d0 and length 2l, which has a small gap in the center of the wire to accomodate a balanced feed using a so-called two-wire transmission line, also known as twin lead or Lecher line. Furthermore, we will assume that the diameter d0 of the wire very small compared to the wavelength d0  λ0. Fig. 4.4 illustrates the linear wire antenna connected to a transmission line. The current distribution along the two-wire transmission line will be sinusoidal, when we neglect radiation and other losses. The wire antenna can now be interpreted as last part of an open-ended 4.4. THIN-WIRE ANTENNAS 73

z

P

r

Je

Generator y 2l

x

Figure 4.4: Thin (lineair) wire antenna of length 2l connected to a two-wire transmission line. The center of the wire is located in the origin of the coordinate system. The wire antenna is directed along the z-axis. two-wire transmission line. The radiation from this folded part of the transmission line is much larger as compared to the spurious radiation from the transmission line itself. This is, of course, our aim, since we want to construct an efficient antenna. Nevertheless, the current distribution along the wire will remain the same (first-order approximation) as the un-folded open ended transmission line, so with the sinusoidal current distribution. The current distribution at the end point of the antenna must be equal to zero. Furthermore, we will assume that the gap at the center of the wire antenna is negligibly small. With all these assumptions, the current distribution

J~e along the wire takes the following form:

J~e = I0δ(x)δ(y) sin [k0(l − |z|)]~ez, (4.77) for −l < z < l. Lateron in this book we will use numerical methods to determine the exact current distribution along the wire. It will be shown that (4.77) is a very accurate approximation of the current. It turns out that (4.77) is a good approximation when the diameter of the wire is smaller than λ0/100. Some examples of the current distribution for various lengths of the wire antenna are illustrated in Fig. 4.5. The electric field in the far-field region of the wire antenna can be determined using expression (4.60)

−k2 e−k0r Z 0 k0(~er,~r0) (4.78) E~ = ~er × ~er × J~e(~r0)e dV0, ω0 4πr V0 74 CHAPTER 4. ANTENNA THEORY

λ /2 3λ /4 λ 3λ /2 0 0 0 0 2λ0

Figure 4.5: Current distribution along a thin wire antenna for various lengths according to (4.77).

Since the current is directed along the z-axis we can rewrite ek0(~er,~r0) into the expression ek0z0 cos θ. Furthermore,

~er × ~er × ~ez = ~eθ sin θ. (4.79)

As a result, we can conclude that the electric field E~ only has one component, being Eθ. The radiated field is linearly polarized and takes the following form

l −k2I e−k0r Z 0 0 k0z0 cos θ Eθ = sin θ sin [k0(l − |z0|)] e dz0, (4.80) ω0 4πr −l

As a first step towards a closed-form solution of (4.80), we will determine the integral by dividing 4.4. THIN-WIRE ANTENNAS 75 it into two parts:

l Z k0z0 cos θ Iz0 = sin [k0(l − |z0|)] e dz0 −l l 0 Z Z k0z0 cos θ k0z0 cos θ = sin [k0(l − z0)] e dz0 + sin [k0(l + z0)] e dz0 0 −l l l Z Z k0z0 cos θ −k0z0 cos θ = sin [k0(l − z0)] e dz0 + sin [k0(l − z0)] e dz0 0 0 (4.81) l Z = 2 sin [k0(l − z0)] cos(k0z0 cos θ)dz0 0 l l Z Z = 2 sin(k0l − k0z0 + k0z0 cos θ)dz0 + sin(k0l − k0z0 − k0z0 cos θ)dz0 0 0

2 [cos(k0l cos θ) − cos(k0l)] = 2 . k0 sin θ Substitution of (4.81) in (4.80) provides:

e−k0r cos(k l cos θ) − cos(k l) E = Z I 0 0 . (4.82) θ 0 0 2πr sin θ

Note that Eθ does not depend on the φ coordinate, which could have been expected as a result of the symmetry. The corresponding magnetic field is now easily found, since in the far-field region we have

1 H~ = ~er × E~ Z0 1 (4.83) = ~er × Eθ~eθ Z0 1 = ~eφEθ. Z0

As a result, only the Hφ component is non zero and is given by

1 Hφ = Eθ Z0 (4.84) I e−k0r cos(k l cos θ) − cos(k l) = 0 0 0 . 2πr sin θ

Note that also Hφ does not depend on the φ coordinate. The Pointing vector (power density in [W/m2]) is now found by using the expression (4.59):

1 2 S~(~r) = |E~ | ~er 2Z0 (4.85) Z |I |2 cos(k l cos θ) − cos(k l)2 = 0 0 0 0 ~e . 8π2r2 sin θ r 76 CHAPTER 4. ANTENNA THEORY

We will now determine the normalized radiation pattern for several wire antennas with varying length. The radiation pattern is found by using expression (2.10).

i. λ0/2 wire antenna.

When 2l = λ0/2 we find that Eθ is equal to

e−k0r cos((π/2) cos θ) E = Z I . (4.86) θ 0 0 2πr sin θ

The radiation pattern of the λ0/2 dipole antenna is shown in Fig. 4.6. Note that the radiation pattern is quite similar to the radiation patters of the electric dipole, which is also shown in 0 the figure. The 3dB beam width of the λ0/2 dipole is 78 , while the beam width of an electric dipole is 900.

90 1 120 60

0.8

0.6 150 30 0.4

0.2

180 0

210 330

240 300

270

Figure 4.6: Radiation pattern versus θ of an λ0/2 wire antenna (full line) and of an electric dipole (dotted line).

ii. λ0 wire antenna.

Now 2l = λ0 and Eθ is given by

e−k0r cos(π cos θ) + 1 E = Z I . (4.87) θ 0 0 2πr sin θ

The corresponding radiation pattern of the λ0 antenna is shown in Fig. 4.7. The 3dB beam width is 470. 4.4. THIN-WIRE ANTENNAS 77

90 1 120 60

0.8

0.6 150 30 0.4

0.2

180 0

210 330

240 300

270

Figure 4.7: Radiation pattern versus θ of an λ0 wire antenna (full line) and of an electric dipole (dotted line).

iii. 3λ0/2 wire antenna.

With 2l = 3λ0/2 we find that Eθ takes the form:

e−k0r cos((3π/2) cos θ) E = Z I . (4.88) θ 0 0 2πr sin θ

The corresponding radiation pattern of the 3λ0/2 antenna is shown in Fig. 4.8. We observe multiple beams in various directions. In addition, the antenna does not radiate energy in several other directions (pattern nulls).

In order to determine the other antenna parameters of the wire antenna, such as the directivity and radiation resistance, we first need to calculate the total radiated power Pt by integrating the power density over a sphere according to (2.17)

ZZ 2 Pt = S~(~r) · ~err dΩ π Z |I |2 Z cos(k l cos θ) − cos(k l)2 = 0 0 2π 0 0 sin θdθ (4.89) 8π2 sin θ 0 Z |I |2 = 0 0 I . 4π p

A general analytic solution of the integral is not easily found. However, in case of an λ0/2-wire antenna we can express the integral Ip in terms of the cosine integral. By applying the coordinate 78 CHAPTER 4. ANTENNA THEORY

90 1 120 60

0.8

0.6 150 30 0.4

0.2

180 0

210 330

240 300

270

Figure 4.8: Radiation pattern versus θ of an 3λ0 wire antenna (full line) and of an electric dipole (dotted line).

transformation u = cos θ and dθ = −du/ sin θ we can write the integral Ip in the following form

π Z cos(k l cos θ) − cos(k l)2 I = 0 0 sin θdθ p sin θ 0 +1" # Z cos2(πu/2) = du 1 − u2 −1 +1" # +1" # 1 Z cos2(πu/2) 1 Z cos2(πu/2) = du + du 2 1 + u 2 1 − u −1 −1 2π" # 2π" # 1 Z cos2((t − π)/2) 1 Z cos2((π − s)/2) = dt + ds (4.90) 2 t 2 s 0 0 2π" # Z cos2((t − π)/2) = dt t 0 2π 1 Z 1 + cos(t − π) = dt 2 t 0 2π 1 Z 1 − cos t = dt, 2 t 0 where we have used the substitutions 1+u = t/π and 1−u = s/π. Furthermore, we have used the fact that cos2(x/2) = (1+cos x)/2. The integral can now be expressed in terms of the well-known 4.4. THIN-WIRE ANTENNAS 79 cosine integral ci(x)

∞ Z cos t ci(x) = − dt. (4.91) t x The cosine integral is well tabulated (for example in the famous book of I. S. Gradsteyn en I. M. Ryzhik, “Table of integrals, series and products”, tabel (8.230), [14]) and is a standard function in Matlab (function cosint). Expression (4.90) now takes the form

2π 1 Z 1 − cos t I = dt p 2 t 0 1 (4.92) = (C + ln(2π) − ci(2π)) 2 2.437 = , 2 where C = 0.577 is Euler’s constant. The total radiation power is now found from:

2 Z0|I0| Pt = Ip 4π (4.93) 1 = 73.1|I |2. 2 0 By using definition (2.16) of the radiation resistance, we easily find that

Ra = 73.1Ω. (4.94)

This suggest that Ra is not frequency dependent. However, this is not true, since we have assumed that the antenna length 2l = λ0/2. Therefore, this antenna is a resonating structure, which implies that the input impedance is strongly frequency dependent. The resonant wire antenna provides a purely resistive load Ra seen at the center of the wire antenna (see Fig. 4.4). In general, the load impedance at the termination of the two-wire transmission line will take a complex form according to Za = Ra + Xa. Only in case of a resonating wire antennas with 2l = λ0/2 is Xa = 0 and a small diameter d0  λ0, we find a purely resistive load impedance. The calculation of the complex input impedance of a wire antenna Za = Ra + Xa outside resonance is more complex and can only be done by using numerical techniques such as the Method of Moments. In chapter 5 we will investigate numerical methods.

Finally, we will determine the directivity of the λ0/2 wire antenna. By using the definition of the directivity function (2.11), we obtain

2 Z0|I0| max(P (θ, φ)) 8π2 D = = 2 = 1.64. (4.95) Pt/4π 73.1|I0| /2 4π

The corresponding directivity expressed in Decibels (10log10D) is DdB = 2.15 dB, which is slightly higher than the directivity of the electric dipole D = 1.5 or DdB = 1.76 dB 80 CHAPTER 4. ANTENNA THEORY

4.5 Wire antenna above a ground plane

Up to now we have assumed that the antenna is located in free space. However, in practise the antenna will often be placed close to objects, e.g. earth surface or in case of printed antennas close to a ground plane of the printed-circuit board (PCB). As a result, the radiation properties and input impedance will change. In this section we will investigate the effect of a perfectly conducting ground plane on the radiation properties of wire antennas. We will assume that the horizontally-oriented resonating wire antenna of length 2l = λ0/2 is located above the conducting ground plane at a height h. The configuration is shown in Fig. 4.9. In order to solve the antenna

wire antenna Je

h Ground plane

x h

Je,image image antenna

Figure 4.9: Thin horizontal wire antenna of length 2l = λ0/2 above a perfectly conducting infinite ground plane. The antenna is situated along the x-axis (y = 0). problem of Fig. 4.9, we can use the so-called image-methods. This method is based on the fact that the tangential component of the electric field E~ vanishes at the perfectly-conducting ground plane. Exactly the same effect can be obtained by removing the conducting plate and placing a second horizontal wire antenna (image antenna) at a distance 2h from the original wire antenna, as illustrated in Fig. 4.9. The current distribution on the image antenna is 1800 out of phase with the current distribution on the original horizontal wire antenna. In this way, we have defined an equivalent problem of the original wire antenna above a ground plane. The equivalent problem can be easily solved by applying the superposition principle and by using the expressions found in section 4.4. Note that we have assumed that the current distribution of the resonant wire antenna is still purely sinusoidal. It is now interesting to investigate the radiation pattern of the horizontal wire antenna above a 4.6. FOLDED DIPOLE 81 ground plane in more detail in the φ = 900-plane. Note that we had assumed that the antenna is located along the x-axis. Based on the symmetry in the y − z plane, the radiation pattern of a wire antenna in free-space would have been independent of the angle θ. The ground plane makes the antenna θ-dependent. Now let E~ o be the radiated electric field from a wire antenna in free-space and E~ i the corresponding radiated electric field of the image-antenna. By applying the superposition principle we obtain the total field in the far-field region as:

E~ tot = E~ o + E~ i

h i k0h cos θ −k0h cos θ (4.96) = E~free e − e

= E~free2 sin(k0h cos θ),

where the vector E~free represents the electric field of a horizontal wire antenna in free space and located at the origin of the coordinate system. The factor 2 sin(k0h cos θ) describes the effect of the ground plane. The corresponding radiation pattern, normalized to an horizontal wire antenna 0 in free-space (so ignoring the factor E~free in (4.96)) is now given for the φ = 90 plane by:

P (θ, φ = 900) Fr(θ) = 0 Pfree(θ, φ = 90 ) (4.97) 2 = 4 sin (k0h cos θ).

The antenna does not radiate energy in the horizontal plane with θ = 900. Fig. 4.10 shows the radiation pattern versus the height h above the ground plane. The directivity of a horizontal wire antenna above a ground plane is larger as compared to a wire antenna in free space. This is directly related to the fact that the antenna does not radiate energy in the half-space below the ground plane. The directivity depends on the height h and can be up to a factor 4 (6 dB) higher as compared to a free-space wire antenna. The beam width is also strongly dependent on 0 0 the height h (See Fig.4.10). In case h = 0.1λ0, the beam width in the φ = 90 -vlak is θHP = 94 .

4.6 Folded dipole

We have previously shown that a basic λ0/2 wire antenna has an input impedance at resonance of approx. 73Ω. A two-wire transmission line (Lecherline or twin-lead) has a somewhat higher t characteristic impedance, for example Z0 = 300Ω. As a result a matching circuit would be required to match the antenna to the transmission line, resulting in additional losses and bandwidth limitations. By modifying the basic λ0/2 wire antenna to the structure of Fig. 4.11 we can realise a much higher input impedance. This antenna structure is called Folded dipole. It can be shown that the input impedance of the λ0/2 folded dipole is about 4 times higher as compared to the normal λ0/2 wire antenna, so Zin ≈ 4 · 73 = 292Ω. The radiation properties of the folded λ0/2 dipole are quite similar to the characteristics of a normal λ0/2 dipole. 82 CHAPTER 4. ANTENNA THEORY

90 1 120 60

0.8

0.6 150 30 0.4

0.2

180 0

210 330

240 300

270

Figure 4.10: Normalized radiation pattern versus θ in the φ = 900 plane of a horizontal wire antenna above a perfectly-conducting infinite metal ground plane. A linear scale is used. With h = 0.1λ0 full line, h = 0.5λ0 dotted line and h = λ0 line with dotts.

4.7 Loop antennas

The loop antenna consists of a circular thin metal wire with radius a. In this section we will assume that the current I0 along the loop is constant, which is only a valid approximation when the radius is very small as compared to the wavelength, i.e a  λ0. For larger loop antennas, this can only be realised by using additional phase-shifters in the loop. Figure 4.12 shows the geometry of the loop antenna. The electric current distribution J~e can now be represented by: q  ~ 2 2 Je(~r0) = I0δ x0 + y0 − a δ(z0)~eφ0 . (4.98)

In chapter 4.2 we found that the far field can be determined with:

−k e−k0r Z H~ = 0 ~e × J~ (~r )ek0(~er,~r0)dV , 4πr r e 0 0 V0 (4.99)

E~ = Z0H~ × ~er.

The integration for the loop antenna is carried out over a circle with radius r0 = a. We will use polar coordinates to facilitate the integration. Furthermore

(~er, ~r0) = ([cos φ sin θ~ex + sin φ sin θ~ey + cos θ~ez], [a cos φ0~ex + a sin φ0~ey]) (4.100)

= a cos(φ − φ0) sin θ. 4.7. LOOP ANTENNAS 83

λ0/2

Two-wire transmission line

Figure 4.11: Folded λ0/2 dipole.

Substituting this in (4.99) provides

2π −k e−k0r Z H~ = 0 (I a)~e × ~e ek0a cos(φ−φ0) sin θdφ . (4.101) 4πr 0 r φ0 0 0

Due to the symmetry of the geometry, we can assume that the fields will not depend on the φ coordinate. Therefore we can use φ = 0 in (4.101). In addition, ~eφ0 = −~ex sin φ0 + ~ey cos φ0. In the φ = 0 plane it is found that ~eφ = ~ey. We can now divide the integral in (4.101) into two parts according to

2π Z ~ k0a cos φ0 sin θ Iφ0 = ~eφ0 e dφ0 0 2π 2π Z Z k0a cos φ0 sin θ k0a cos φ0 sin θ = −~ex sin φ0e dφ0 + ~ey cos φ0e dφ0 (4.102) 0 0 2π Z k0a cos φ0 sin θ = ~ey cos φ0e dφ0. 0

The first integral is equal to zero, since the integrand is an odd function with respect to φ0. The 84 CHAPTER 4. ANTENNA THEORY

z P

θ

r

I0

a y φ I0

x

Figure 4.12: Loop antenna consisting of a circular thin metal wire with radius a carrying a constant current I0. The antenna is situated in the x − y-plane. remaining integral can be analytically solved: 2π Z ~ k0a cos φ0 sin θ Iφ0 = ~ey cos φ0e dφ0 0 π Z k0a cos φ0 sin θ = 2~ey cos φ0e dφ0 0 π/2 Z −k0a sin φ0 sin θ = −2~ey sin φ0e dφ0 −π/2 π/2 Z   (4.103) k0a sin φ0 sin θ −k0a sin φ0 sin θ = 2~ey sin φ0 e − e dφ0 0 π/2 Z = 4~ey sin φ0 sin(k0a sin φ0 sin θ)dφ0 0

= 2πJ1(k0a sin θ)~ey

= 2πJ1(k0a sin θ)~eφ where J1(z) is the Bessel function of the first kind with order 1 and argument z. Bessel functions 4.7. LOOP ANTENNAS 85 are well tabulated (e.g. [14]) and are available in Matlab and other mathematical tools. Finally, we obtain the following expression for the magnetic field in the far-field region using ~er ×~eφ = −~eθ:

k aI e−k0r H~ = H ~e = − 0 0 J (k a sin θ). (4.104) θ θ 2r 1 0

Since ~eθ × ~er = −~eφ, we can conclude that the corresponding electric field will only have a φ component:

k aZ I e−k0r E~ = E ~e = 0 0 0 J (k a sin θ). (4.105) φ φ 2r 1 0

4.7.1 Small loop antennas: magnetic dipole A small loop antenna, also known as magnetic dipole, consists of a circular metal wire with radius a  λ0 and with a constant current I0 along the loop. We can use expression 4.104 and 4.105 to determine the far fields, where we can approximate the Bessel function J1(z) for small argument z according to z J (z) ≈ . (4.106) 1 2 By substituting this in (4.104) and (4.105), we find that the far fields are given by:

2 2 −k0r k0(πa I0)e Hθ = − sin θ 4πr (4.107) k2Z (πa2I )e−k0r E = 0 0 0 sin θ. φ 4πr 2 The factor πa I0 is also known as the ”transmitter-moment” and is the product of the current I0 and the surface of the loop. Usually it is denoted by m with

2 m = πa I0. (4.108) There is a strong analogy between the fields radiated by a magnetic dipole according to (4.107) and the fields from the electric dipole, given by (4.68) and (4.69). The radiation patterns of the magnetic and electric dipole are exactly the same. Now let us assume that both an electric as well as an magnetic dipole are located in the origin of our coordinate system. Furthermore, assume that the electric dipole is oriented along the z-axis and the magnetic dipole is situated in the x-y-plane. When we choose the dimensions of both dipoles and current to fulfill the relation

p = −mk0, (4.109) the corresponding electromagnetic field generated by these two sources will fullfill the following relation

E~ = Z0H.~ (4.110) The minus sign in (4.109) indicates that the currents in the electric and magnetic dipoles are 1800 out of phase. When the currents are in phase, i.e. p = mk0, we find the following form of the radiated electromagnetic field:

E~ = −Z0H.~ (4.111) 86 CHAPTER 4. ANTENNA THEORY

Electromagnetic fields satisfying (4.110) or (4.111) have the unique property that the radiated far fields are circularly polarized in all directions. Using (4.110) we find that

Eφ = Eθ, (4.112)

for all directions (θ, φ). Therefore, we satisfy the relation between Eθ en Eφ as described in section 2.9 for a perfectly left-handed circularly-polarized field. Fields that satisfy (4.111) are right-handed circularly polarized for all directions (θ, φ).

Exercise Calculate the directivity of a magnetic dipole.

4.7.2 Large loop antennes

The electromagnetic fields in the far-field region of a loop antenna with uniform current distribu- tion along the loop was already determined in (4.104) and (4.105). We can use these expressions to determine the radiation pattern, resulting in

P (θ) F (θ) = max(P (θ)) (4.113) 2 |J1(k0a sin θ)| = 2 . max(|J1(k0a sin θ)| )

The shape of the pattern will be similar to the behavior of the Bessel function J1(z). Fig. 4.13 shows the normalized radiation pattern of a loop antenna with radius a = λ0/20, a = λ0/2, a = 2λ0, respectively. Note that the direction of maximum radiation moves towards θ = 0 with increasing loop radius (check Fig. 4.12) .

4.8 Magnetic sources

In order to be able to introduce aperture antennas, it is useful to extend Maxwell’s equations, given by (4.1). We will do this by introducing magnetic currents and associated magnetic charge distributions. Note that both magnetic currents as well as magnetic charges do not exist in the real-world: both are non-physical quantities.. However, it will help us to introduce several new electromagnetic concepts in an elegant way.

For that purpose, we will introduce the magnetic current distribution J~m(~r) and the magnetic charge density ρm(~r). When we assume that there are no electric currents J~e(~r) = ~0 and electric 4.8. MAGNETIC SOURCES 87

90 1 120 60 0.8

0.6 150 30 0.4

0.2

180 0

210 330

240 300

270

Figure 4.13: Normalized radiation pattern versus θ of a loop antenna with uniform current distri- bution for various loop dimensions. With a = λ0/20 full line, a = λ0/2 dotted line and a = 2λ0 line with dotts. charges ρe(~r) = 0, Maxwell’s equations take the following form in the space-frequency domain:

∇ × E~ (~r) = −ωµ0H~ (~r) − J~m(~r),

∇ × H~ (~r) = ω0E~ (~r), (4.114) ρ (~r) ∇ · H~ (~r) = m , µ0

∇ · E~ (~r) = 0.

By combining the first and third equation from (4.114) we observe that the following continuity equation yields:

(4.115) ∇ · J~m(~r) + ωρm(~r) = 0.

The solution of these equations related to magnetic sources can be found by using a similar approach as used in section 4.1 for the case of electric sources. As a first step, we will introduce a vector potential, in this case the electric vector potential A~m, using the fourth equation in (4.114). As a result, we find that 1 E~ = − ∇ × A~m. (4.116) 0 88 CHAPTER 4. ANTENNA THEORY

This gives ∇ × H~ = −ω∇ × A~m. We can now write H~ in the form:

H~ = −ωA~m − ∇φm, (4.117) where φm is (yet) an arbitrarily scalar function. By substituting this relation into the first equation of (4.114) and using the Lorentz-gauge according to

∇ · A~m = −ω0µ0φm, (4.118) leads towards the well-known Helmholtz equation for the vector potential A~m, given by

2 ~ 2 ~ ~ ∇ Am + k0Am = −0Jm. (4.119)

A similar expression can be found for the scalar potental φm directly from the Lorentz-gauge that we used. In section 4.1 we solved Helmholtz equation for an arbitrarily electric current distribution

J~e by first looking for the solution for an elementary point source. The general solution was then found by applying the superposition concept, since any electric current distribution J~e can be written as a continuous superposition of point sources. By applying the same strategy here, we

find that the electric vector potential A~m due to an arbitrarily magnetic current distribution J~m takes the following form:

−k |~r−~r | 0 Z e 0 0 A~m(~r) = J~m(~r0) dV0, (4.120) 4π V0 |~r − ~r0| where we have assumed that the observation point P , indicated by the vector ~r, is located outside the source region V0. The electromagnetic fields outside the source region V0 are now found by

1 E~ = − ∇ × A~m, 0 (4.121) 1 H~ = ∇ × ∇ × A~m. ω0µ0 Similar to section 4.2, we can determine the electromagnetic fields far away from the sources (far-field region) by using several approximations. The fields in the far-field region than take the following form:

k e−k0r Z 0 k0(~er·~r0) E~ = ~er × J~m(~r0)e dV0, 4πr V0 −k2 e−k0r Z 0 k0(~er·~r0) (4.122) H~ = ~er × ~er × J~m(~r0)e dV0, ωµ0 4πr V0

E~ = Z0H~ × ~er,

We again can conclude that the fields in the far-field region locally behave as TEM-waves (E-

field perpendicular to H and perpendicular to the propagation direction ~er. The strong analogy between magnetic and electric sources is known in literature as the duality concept. Always keep in mind that magnetic currents and charges are mathematical quantities that cannot exist in nature. 4.9. LORENTZ-LAMOR THEOREM 89

We have now introduced all tools to determine the radiated electromagnetic fields from a general source, consisting of both electric and (fictive) magnetic currents and charges. We can do this by using expressions (4.60) and (4.122) and by applying the superposition principle.

Exercise In the origin of our coordinate system a magnetic current distribution exist with the following properties:

J~m(~r0) = I0mlδ(x)δ(y)δ(z)~ez, (4.123)

where I0m is the total magnetic current. Show that the radiated electromagnetic fields due to this source is identical to the fields due to the magnetic dipole (small loop antenna) as discussed in section 4.7.1.

4.9 Lorentz-Lamor theorem

In order to analyse aperture antenna, it is convenient to introduce the Lorentz-Lamor theorem. This concept defines the electromagnetic relation between two situations A and B. Later in this section we will show that we can apply this theorem to replace a particular antenna problem, a ~a represented by a source region V with electric and magnetic (volume-) current distributions Je ~a en Jm, by an equivalent and simpler antenna problem. In literature this concept is also known as the Equivalence principle [15]. Figure 4.14 illustrates source region V a bounded by the surface a ~a ~a ~ a ~ a a S . The sources Je en Jm generate an electromagnetic field E and H outside source region V . It will be shown that the same fields outside V a can also be generated by an equivalent electric ~a a surface-current distribution Jes on S and an equivalent magnetic surface-current distribution ~a a Jms on S where

~a ~ a Jes = ~en × H , (4.124) ~a ~ a Jms = E × ~en,

a a where ~en is the normal vector on surface S . The equivalent sources (4.124) produce within V an zero electromagnetic field and outside V a an electromagnetic field equal to the original problem.

Intermezzo: derivation of Lorentz-Lamor theorem The derivation of the Lorentz-Lamor theorem provides interesting insight into the relation and electromagnetic interaction between two source regions. As a first step, we will formulate Maxwell’s equations in the space-frequency domain for situation A where the fields E~ a and H~ a are generated 90 CHAPTER 4. ANTENNA THEORY

e S n V

a a J e J m

V S a   b S b b a J e J m

Vb

Figure 4.14: Source region V a enclosed by the surface Sa and source region V b enclosed by surface Sb. Both regions are located within volume V with corresponding surface S. Within source region a ~a ~a V both electric and magnetic current distributions Je and Jm exist and the corresponding charge a a b distributions ρe en ρm. Similar for source region V . only by the sources in volume V a (see also Fig. 4.14):

~ a ~ a ~a ∇ × E = −ωµ0H − Jm,

~ a ~ a ~a ∇ × H = ω0E + Je , (4.125) ρa ∇ · H~ a = m , µ0 ρa ∇ · E~ a = e . 0 ~b ~b b In situation B, a source distribution Je and Jm exist within the volume V together with the b b corresponding charge distributions ρe and ρm. The fields generated by these sources (so without the Situation A sources) comply tot the following Maxwell equations :

~ b ~ b ~b ∇ × E = −ωµ0H − Jm,

~ b ~ b ~b ∇ × H = ω0E + Je , (4.126) ρb ∇ · H~ b = m , µ0 ρb ∇ · E~ b = e . 0 4.9. LORENTZ-LAMOR THEOREM 91

The divergence of the vector funktion E~ a × H~ b − E~ b × H~ a is: h i h i h i ∇ · E~ a × H~ b − E~ b × H~ a = ∇ · E~ a × H~ b − ∇ · E~ b × H~ a . (4.127)

The first term of the right-hand side of equation (4.127) can be written as

h i ∇ · E~ a × H~ b = H~ b · ∇ × E~ a − E~ a · ∇ × H~ b

~ b  ~ a ~a  ~ a  ~ b ~b (4.128) = H · −ωµ0H − Jm − E · ω0E + Je

~ b ~ a ~ b ~a ~ a ~ b ~ a ~b = −ωµ0H · H − H · Jm − ω0E · E − E · Je , where we have used the vector identity ∇ · (A~ × B~ ) = B~ · ∇ × A~ − A~ · ∇ × B~ . In a similar way, we can determine the second term in the right-hand side of equation (4.127) by exchange of a and b in (4.128):

h ~ b ~ ai ~ a ~ b ~ a ~b ~ b ~ a ~ b ~a (4.129) ∇ · E × H = −ωµ0H · H − H · Jm − ω0E · E − E · Je .

Combining (4.128) and (4.129) provides the following result

h ~ a ~ b ~ b ~ ai h ~ b ~a ~ b ~a i h ~ a ~b ~ a ~b i ∇ · E × H − E × H = E · Je − H · Jm − E · Je − H · Jm . (4.130)

Applying Gauss theorem according to Z Z ∇ · AdV~ = A~ · ~endS (4.131) V S on the volume V enclosed by the surface S (see Fig. 4.14) provides Z Z h ~ a ~ b ~ b ~ ai h ~ b ~a ~ b ~a i h ~ a ~b ~ a ~b i E × H − E × H · ~endS = E · Je − H · Jm − E · Je − H · Jm dV. (4.132) S V This relation is known as Lorentz’ theorem. In order to derive the Lorentz-Lamor theorem, we will apply Lorentz’ theorem on several special cases.

i. Volume V encloses the entire free-space V∞. The corresponding surface S is now a sphere S∞ with infinite large radius. The sources (E~ a, H~ a) and (E~ b, H~ b) are located within a finite region. At large distances from these sources we have, according to (4.60) and (4.122), the following relations:

a 1 a H~ = ~er × E~ , Z0 (4.133) b 1 b H~ = ~er × E~ . Z0 92 CHAPTER 4. ANTENNA THEORY

Furthermore, for a point with location vector ~r on the sphere S∞ we know that ~er = ~en. Therefore, the integrand of the left-hand side of expression (4.132) can be written in the following form

h a b b ai h ai b h bi a E~ × H~ − E~ × H~ · ~en = ~en × E~ · H~ − ~en × E~ · H~

h ai b h bi a = ~er × E~ · H~ − ~er × E~ · H~ (4.134) a b b a = Z0H~ · H~ − Z0H~ · H~

= 0

The surface integral in (4.132) will vanish when S = S∞. Therefore, expression (4.132) can be reduced to

Z Z h ~ b ~a ~ b ~a i h ~ a ~b ~ a ~b i E · Je − H · Jm dV = E · Je − H · Jm dV. (4.135)

V∞ V∞

Although the integration in (4.135) is performed over the entire volume V∞, it is evident that the integration can be limited to the (finite) source volume.

ii. Source distribution B is an electric dipole. Assume that our source distribution is an electric dipole with dipole moment p = 1 and

direction ~eb located at ~r outside the source regions. This is illustrated in Fig. 4.15. We now

S V eb  a r - r Jre ()0 Q 0 P a Jrm ()0

r0 r

O

Figure 4.15: Source distribution B is an electric dipole with directed towards ~eb with dipole moment p = 1. 4.9. LORENTZ-LAMOR THEOREM 93

obtain

~b Jm = 0, (4.136) ~b Je = ~ebδ(~r − ~r0),

where the vector ~r is fixed and ~r0 variable. When using this relation in (4.135) we find that

Z Z ~ a h ~ b ~a ~ b ~a i ~eb · E (~r0)δ(~r − ~r0)dV0 = E (~r0) · Je (~r0) − H (~r0) · Jm(~r0) dV0, (4.137) a V∞ V

or

Z ~ a h ~ b ~a ~ b ~a i ~eb · E (~r) = E (~r0) · Je (~r0) − H (~r0) · Jm(~r0) dV0. (4.138) V a

In the special case that we only have electric sources within V a, expression (4.138) takes the form:

Z ~ a ~ b ~a ~eb · E (~r) = E (~r0) · Je (~r0)dV0. (4.139) V a

It appears that the electric field E~ a(~r) due to source distribution A can be found by deter- mining the electromagnetic fields of an electric dipole with p = 1 located at ~r with direction

~eb and substituting it in the volume integral. By choosing the direction of the dipole ~eb to be a ~ex, ~ey en ~ez, respectively, we can determine the Cartesian components of E~ (~r). In section 4.1 we have determined the field generated by an electric dipole. Using relation (4.22) and (4.30), we can rewrite (4.139) in the following form

Z ~ a 1 ~a ~ ~eb · E (~r) = Je (~r0) · ∇r0 × ∇r0 × Ae(~r, ~r0)dV0, (4.140) ω0µ0 V a

with

−k |~r−~r | µ0 e 0 0 A~e(~r) = ~eb . (4.141) 4π |~r − ~r0|

Now introduce the variableϕ according to

1 e−k0|~r−~r0| ϕ(~r, ~r0) = . (4.142) 4π |~r − ~r0| 94 CHAPTER 4. ANTENNA THEORY

We can rewrite (4.140) into:

Z ~ a 1 ~a ~eb · E (~r) = Je (~r0) · ∇r0 × ∇r0 × ~ebϕdV0 ω0 V a Z 1 ~a = Je (~r0) · ∇r × ∇r × ~ebϕdV0 ω0 V a (4.143) Z 1 ~a = ~eb · ∇r × ∇r × Je (~r0)ϕdV0 ω0 V a Z 1 ~a = ~eb · ∇r × ∇r × Je (~r0)ϕdV0, ω0 V a

where we have used the relation

∇r0 × ∇r0 × ~ebϕ = ∇r × ∇r × ~ebϕ, (4.144)

~c · ∇ × ∇ × ~ebϕ = ~eb · ∇ × ∇ × ~cϕ.

Note that expression (4.143) is identical to (4.22) in combination with (4.30). We can deter- a mine H~ in a similar way by using an unit magnetic dipole directed along ~eb and located at ~a ~ ~r, while Je = 0.

iii. All sources in volume V . ~a ~a ~b ~b Consider a volume in which the sources Je , Jm, Je and Jm and the corresponding charge distributions exist, see Fig. 4.16. Outside V , which is the volume V2 = V∞ − V , no sources exist. We will again apply Lorentz’ theorem (4.132) on volume V2 = V∞ − V . This volume is bounded by the surface S2 and by S∞. Since there are no sources within V2, the volume integral in (4.132) is equal zero. Therefore, we find that

Z h a b b ai E~ × H~ − E~ × H~ · ~endS = 0, (4.145)

S2+S∞

in which ~en is the unit vector which is directed outwards from V2. Based on the our findings in case i), the contribution over S∞ will be zero. As a result (4.145) will take the form:

Z h a b b ai E~ × H~ − E~ × H~ · ~endS = 0. (4.146)

S2

iv. Source distribution A in volume V and source distribution B in volume V1. ~a ~a Consider the volume V in which the sources Je , Jm and corresponding charge distributions ~b ~b exist. Outside V , in V1, we find the sources Je en Jm with corresponding charge distributions (see Fig. 4.17). Outside V + V1 no sources exist. By again applying Lorentz’ theorem (4.132) 4.9. LORENTZ-LAMOR THEOREM 95

V

a a J e J m

b b J e J m

S2

VVV2 =∞ −

Figure 4.16: All sources are located within volume V .

en

V

a a V1 J e J m

b b J e J m

S2

Figure 4.17: Source distribution A is located in volume V and source distribution B is located in volume V1.

on volume V1 provides:

Z Z h ~ a ~ b ~ b ~ ai h ~ a ~b ~ a ~b i − E × H − E × H · ~endS = − E · Je − H · Jm dV, (4.147)

S V1

where we have used relation (4.146), while the minus-sign in front of the surface integral

accounts for the fact that ~en is directed inwards V1 in stead of outwards V1. Assume that 96 CHAPTER 4. ANTENNA THEORY

source distribution B consists of an electric dipole with direction ~eb and dipole moment p = 1. ~b Furthermore assume that Jm = 0. This is the same situation as used in case ii). From (4.147) now follows that Z a h a bi h b ai ~eb · E~ (~r) = E~ × H~ · ~en − E~ × H~ · ~endS. (4.148) S

Using vector relation (A~ × B~ ) · C~ = −(A~ × C~ ) · B~ finally provides Z a h ai b h a i b ~eb · E~ (~r) = ~en × H~ · E~ − E~ × ~en · H~ dS. (4.149) S

With these four special cases, we can make our final last step in the derivation of the Lorentz- Lamor theorem. For that purpose, we should compare expression (4.149) and (4.138). Both expressions are derived for the same situation as shown in Fig. 4.15. Therefore, we can formulate the following important conclusion:

~a ~a The sources Je and Jm and the corresponding charge distributions within V generate outside V an electromagnetic field E~ a, H~ a. The volume V is enclosed by surface S. The fields E~ a, H~ a ~a outside V can also be assumed to be generated by an electric surface current distribution Jes on ~a S and an magnetic surface current density Jms op S, where

~a ~ a Jes = ~en × H , (4.150) ~a ~ a Jms = E × ~en.

These equivalent sources generate inside volume V an electromagnetic field equal to zero.

End intermezzo: derivation of Lorentz-Lamor theorem

We can further derive the expression for the electric field at location ~r outside the source region V (4.149) by applying the derived expressions for the dipole fields E~ b en H~ b. From (4.30) we find that

~ b 1 E (~r0) = ∇r0 × ∇r0 × ~ebϕ, ω0 (4.151) ~ b H (~r0) = ∇r0 × ~ebϕ, where ϕ is defined in (4.142). Substitution of(4.151) in (4.149) provides Z Z ~ a 1 h ~ a i h ~ a i ~eb · E (~r) = ~en × H (~r0) · ∇r0 × ∇r0 ×~ebϕdS − E (~r0) × ~en · ∇r0 ×~ebϕdS.(4.152) ω0 S S 4.9. LORENTZ-LAMOR THEOREM 97

By using the vector relations provided in (4.144) and relation ∇r0 × ~ebϕ = −∇rϕ × ~eb finally provides Z Z a h a i 1 h a i E~ (~r) = ∇r × ~en × E~ (~r0) ϕdS + ∇r × ∇r × ~en × H~ (~r0) ϕdS. (4.153) ω0 S S with

1 e−k0|~r−~r0| ϕ(~r, ~r0) = . (4.154) 4π |~r − ~r0|

The corresponding magnetic field H~ a(~r) outside source region V van be found by placing a magnetic dipole of unit strength and direction ~eb at the location ~r. From (4.147) we than obtain

Z a h a bi h b ai  ~eb · H~ (~r) = − E~ × H~ · ~en − E~ × H~ · ~en dS S (4.155) Z h ai b h ai b = − ~en × E~ · H~ + ~en × H~ · E~ dS. S

The (magnetic) dipole fields E~ b and H~ b have already been derived in section (4.8) and are given by (4.156). This results in

~ b E (~r0) = −∇r0 × ~ebϕ, (4.156) ~ b 1 H (~r0) = ∇r0 × ∇r0 × ~ebϕ. ωµ0 Substitution of (4.156) in (4.155) results in Z Z ~ a h ~ a i 1 h ~ a i ~eb · H (~r) = ~en × H (~r0) · ∇r0 ×~ebϕdS − ~en × E (~r0) · ∇r0 × ∇r0 ×~ebϕdS.(4.157) ωµ0 S S By again applying vector relations we can rewrite (4.157) in the following form Z Z a h a i 1 h a i H~ (~r) = ∇r × ~en × H~ (~r0) ϕdS − ∇r × ∇r × ~en × E~ (~r0) ϕdS. (4.158) ωµ0 S S

Summary

Assume that the sources J~e, J~m and corresponding charge distributions exist within volume V , enclosed by the surface S, The generated electromagnetic field outside V is then found by:

Z h i 1 Z h i E~ (~r) = ∇r × ~en × E~ (~r0) ϕdS + ∇r × ∇r × ~en × H~ (~r0) ϕdS, ω0 S S (4.159) Z h i 1 Z h i H~ (~r) = ∇r × ~en × H~ (~r0) ϕdS − ∇r × ∇r × ~en × E~ (~r0) ϕdS, ωµ0 S S 98 CHAPTER 4. ANTENNA THEORY where J~es = ~en × H~ and J~ms = E~ × ~en are the equivalent electric and magnetic surface current distribution on the surface S, respectively. When ~r is within volume V we find that

E~ (~r) = ~0, (4.160) H~ (~r) = ~0.

This is the Lorentz-Lamor theorem also referred to as equivalence principle

It is now fairly straightforward to derive expressions for the electromagnetic fields far away from the sources, in the so-called far-field region. The integrand in (4.159) consists of products of vector functions and scalar functions, where the vector function does not depend on ~r, while the operator ∇r× operates only on ~r. Formulas with such a structure have been discussed before in section 4.2. We concluded that for far-field points ~r the following relations can be used:

∇r× → −k0~er×, (4.161) 2 ∇r × ∇r× → (−k0) ~er × ~er × .

Furthermore, in the far-field region the following approximation can be used:

e−k0|~r−~r0| e−k0r ≈ ek0(~r0·~er). (4.162) |~r − ~r0| r Substitution of (4.161) and (4.162) in (4.159) provides the following result

−k e−k0r Z h i h i E~ (~r) = 0 ~e × ~e × E~ (~r ) − ~e × ~e × Z H~ (~r ) ek0(~r0·~er)dS, 4πr r n 0 r n 0 0 S (4.163) −k e−k0r Z h i h i Z H~ (~r) = 0 ~e × ~e × Z H~ (~r ) + ~e × ~e × E~ (~r ) ek0(~r0·~er)dS. 0 4πr r n 0 0 r n 0 S This can also be written in a more compact form:

−k r −k0e 0 E~ (~r) = F~a(~er), 4πr (4.164) −k e−k0r Z H~ (~r) = 0 ~e × F~ (~e ). 0 4πr r a r with

Z h i h i k0(~r0·~er) F~a(~er) = ~er × ~en × E~ (~r0) − ~er × ~en × Z0H~ (~r0) e dS. (4.165) S

The subscript a is used to indicate that the function Fa is found by integrating the fields on the antenna aperture (= surface S). Note that the following relation also holds in the far-field:

E~ = Z0H~ × ~er. (4.166) 4.10. HORN ANTENNAS 99

The formulas (4.164) can now be used to determine the radiation properties of arbitrarily-shaped aperture antennas. A proper choice of the surface S is very important. In the next sections we will show how S can be chosen in a convenient way for some well-known aperture antenna concept. In this way, we are able to find an analytic expression of the radiated fields.

4.10 Horn antennas

A well known example of an aperture antenna is the horn antenna as illustrated in 4.18. Horn antennas are available with rectangular, square and circular apertures and are usually fed from a waveguide. Fig. 4.18 also shows how we could select the closed surface S with the equivalent sources in order to apply the Lorentz-Lamor concept as discussed in the previous section. S consists of two part, namely Sa and Sc. Sa is the actual radiating aperture of the horn antenna, while Sc contains the outer (metallic) surface of the horn antenna and generator. Therefore, all sources are enclosed by the surface S. We can now make the following assumptions:

• The outer surface of the antenna and the generator are perfectly electric conductors (PEC).

This implies that the tangential electric field along this surface is zero: ~en × E~ = ~0 on Sc.

• The (secondary) currents along the outer surface of the antenna and generator are very

small and can be ignored in our analysis, i.e. ~en × H~ = ~0 op Sc.

The integration in (4.163) can now be limited to Sa. As a first-order approximation we will

generator waveguide horn

Sa Sc

Figure 4.18: Example of horn antennas and choice of the enclosed surface S as Sa + Sc. assume that the aperture fields are uniformly distributed in the x-y plane, as indicated in Fig. 4.19. Note that the derivation for the case of a more realistic aperture distribution is quite similar, although the mathematics are more complex or even require a numerical integration. The fields 100 CHAPTER 4. ANTENNA THEORY

b

a

~ E0 a a Figure 4.19: Rectangular uniform aperture with vecEa = E0~ey and Ha = − ~ex for − 2 < x < 2 Z0 b b and − 2 < y < 2 . of the uniformly illuminated aperture can be described as:

E~a = E0~ey, (4.167) E0 H~ a = − ~ex, Z0

a a b b valid for the rectangular region − 2 < x < 2 and − 2 < y < 2 . Furthermore, E0 is a constant and we have assumed that the characteristic impedance is equal to Z0. The equivalent electric and magnetic surface currents on Sa are now given by

J~m = E~a × ~en = E0~ey × ~ez = E0~ex, (4.168) E0 E0 J~e = ~en × H~ a = − ~ez × ~ex = − ~ey. Z0 Z0

We can calculate the electric field in the far-field region by applying expression (4.163) on aperture

S = Sa. We now find

a/2 b/2 −k e−k0r Z Z   E~ (~r) = 0 ~e × −J~ − Z ~e × J~ ek0(~r0·~er)dx dy 4πr r ms 0 r es 0 0 −a/2 −b/2 (4.169) k e−k0r = 0 (L + L ) , 4πr m e 4.10. HORN ANTENNAS 101 where the integrals Lm and Le are given by a/2 b/2 Z Z k0(~r0·~er) Lm = ~er × J~ms(~r0)e dx0dy0, −a/2 −b/2 a/2 b/2 (4.170) Z Z k0(~r0·~er) Le = Z0 ~er × ~er × J~es(~r0)e dx0dy0. −a/2 −b/2 We can determine the integrals easily by writing the in- and outer-products in (4.170) in terms of spherical coordinates (r, θ, φ):

~er × ~ex = ~er × (~eθ cos θ cos φ − ~eφ sin φ) = ~eφ cos θ cos φ + ~eθ sin φ,

~er × ~er × ~ey = −~eθ cos θ sin φ − ~eφ cos φ, (4.171)

(~r0 · ~er) = (~exx0 + ~eyy0) · (~ex sin θ cos φ + ~ey sin θ sin φ + ~ez cos θ)

= x0 sin θ cos φ + y0 sin θ sin φ = x0u + y0v, with u = sin θ cos φ and v = sin θ sin φ. Substitution of (4.171) and (4.168) in (4.170) provides for the first integral Lm a/2 b/2 Z Z k0(x0u+y0v) Lm = E0 (~eφ cos θ cos φ + ~eθ sin φ) e dx0dy0, −a/2 −b/2 (4.172)     sin(k0au/2) sin(k0bv/2) = abE0 (~eφ cos θ cos φ + ~eθ sin φ) , k0au/2 k0bv/2 where we have used the well-known integral

a/2 Z sin(βa/2) eβxdx = a . (4.173) βa/2 −a/2

Similarly, we can determine the integral Le as

    sin(k0au/2) sin(k0bv/2) (4.174) Le = abE0 (~eθ cos θ sin φ + ~eφ cos φ) . k0au/2 k0bv/2 Finally, by substituting (4.174) and (4.172) in (4.169) we obtain the following expression for the electric field in the far-field region of the horn antenna

k e−k0r E~ (~r) = 0 (L + L ) 4πr m e −k0r     k0abE0e sin(k0au/2) sin(k0bv/2) (4.175) = [~eφ cos φ(cos θ + 1) + ~eθ sin φ(cos θ + 1)] 4πr k0au/2 k0bv/2

= Eθ~eθ + Eφ~eφ. 102 CHAPTER 4. ANTENNA THEORY

We can observe that E~ has two components, Eθ~eθ and Eφ~eφ. In the principle planes of the antenna, i.e. the φ = 0 plane and φ = π/2 plane, one of both components is equal to zero. The magnetic field H~ in the far-field region is easily found by using the far-field relation E~ = Z0H~ ×~er. The radiation pattern is determined according to the approach in chapter 2 by using relation (2.8) up to (2.10). The resulting normalized radiation pattern then takes the following form:

2 2 |Eθ(θ, φ)| + |Eφ(θ, φ)| F (θ, φ) = 2 2 . (4.176) |Eθ(0, 0)| + |Eφ(0, 0)| Fig. 4.20 shows the three-dimensional radiation pattern in [dB] for a uniform aperture with dimensions of a = 4λ0 and b = 2λ0. The corresponding cross section in the φ = 0-plane (v = 0) is illustrated in Fig. 4.21. The first sidelobe has a peak at −13.26 dB w.r.t. the main lobe. The 0 3-dB beam width is θHP = 12.4 .

Figure 4.20: Three-dimensional normalized radiation pattern versus (θ, φ) of a uniform aperture with dimensions a = 4λ0 and b = 2λ0. All values are in dB.

4.11 Reflector antennas

Reflector antennas can take many shapes, e.g. parabolic, spherical or cylindrical. The parabolic reflector antennas is the most commonly used type. Reflector antennas provide a large antenna gain and associated effective antenna aperture, which makes them very suited for long-range wireless applications, such as satellite communication, point-to-point (PtP) backhauling and long- range radar. The parabolic reflector antenna is fed by a small antenna element(de feed), such as a dipole antenna or horn antenna. The feed is placed in the focal point of the reflector and illuminated the reflector. As a result of this, electric current are generated on the metal reflector surface. These currents generate the scattered field of the reflector. Fig. 4.22 shows an illustration of the parabolic reflector and feed antenna. The electromagnetic field in the far-field region can be determined using equation (4.163). In this case, we will choose S = Sa + Sc as indicated in Fig. 4.22a. The sources of the scattered field are now all located within the surface S. We will 4.11. REFLECTOR ANTENNAS 103

Uniform rectangular aperture φ=0 plane, a=4λ and b=2λ 0 0 0

−5

−10

−15

−20

−25 Normalised radiation pattern [dB]

−30

−35

−40 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 4.21: Cross section in the φ = 0-plane of the normalized radiation pattern versus θ of a uniform aperture with dimensions a = 4λ0 and b = 2λ0. All values are in dB. The 3dB beam 0 width in this plane is θHP = 2arcsin(0.433λ0/a) = 12.4

Sc Sc Sa Sa

F F F Feed antenna Feed antenna

Parabolic reflector Parabolic reflector

Figure 4.22: Parabolic reflector (cross section) with feed placed in the focal point. (a) The choice of the surface Sa according to the aperture method, (b) surface Sa is selected according to the current-integration method.. 104 CHAPTER 4. ANTENNA THEORY assume that the reflector is made of a perfectly-conducting metallic material. Furthermore, we will ignore the currents flowing on the back side of the reflector. As a result

~en × E~ = ~0 op Sc, (4.177)

~en × H~ = ~0 op Sc.

Therefore, we can limit the integration in (4.163) to the aperture Sa. The radiation pattern of the feed antenna located in the focal point F determines the field distribution over the aperture plane Sa. This method is known as the aperture method. However, it is also possible to select the surface S as indicated in Fig. 4.22b. In this case we find that ~en × E~ = ~0 on Sa, and as a result (4.163) reduces to

k e−k0r Z h i E~ (~r) = 0 ~e × ~e × ~e × Z H~ (~r ) ek0(~r0·~er)dS, (4.178) 4πr r r n 0 a 0 S

The magnetic field in the aperture H~ a on Sa can be easily determined from the radiation charac- teristics of the feed antenna located in the focal point F . This method is known as the current integration method. In this section we will use the apeture method to determine the radiation characteristics of a parabolic reflector antenna. Now consider the configuration of Fig. 4.23. The reflector has a diameter D = 2a. A feed antenna is located in the focal point F . Since the feed antenna is relative small (typically 0.5λ0 − 4λ0), a point A on the reflector will be located in the far-field region of the feed antenna, that is the distance between the feed and the reflector is larger than 2L2 , with L the largest dimension of the feed antenna. In the far-field region of the feed antenna, λ0 the equi-phase phase front is spherical. Perpendicular to these planes, we can define so-called rays directed along the Pointing vector. Now assume that a ray that originates from the feed hits the reflector at point A on the reflector surface, as indicated in Fig. 4.23. We will assume that the reflector at point A is locally flat. Furthermore, the incident wave at point A is locally flat as well, represented as a plane wave. As a result, we can use the well-known reflection rules which yield for a plane wave incident on a large and perfectly-conduction flat plate. In this way we can determine the field in point B, see Fig. 4.23. In this way, we can determine the fields in the aperture Sa. The radiation characteristics of the feed antenna have a very large impact on the final radiation characteristics of the reflector antenna. In many applications the feed is designed in such a way that the edges of the reflector are less illuminated as compared to the center region. This is called amplitude tapering. For the sake of simplicity, we will first assume that the feed antenna provides a uniform illumi- nation of the aperture Sa. Now let the incident electric field E~i coincide with the x-direction and the magnetic field H~ i with the y-direction, as illustrated in Fig. 4.23. In the circular-symmetric aperture, we then have the following field distribution:

E~a = E0~ex, (4.179)

Z0H~ a = E0~ey, 4.11. REFLECTOR ANTENNAS 105

A B

x

E D y Feed antenna

ZoH

Sa Equi-phase plane

Aperture Parabolic reflector

Figure 4.23: Circularly-symmetric parabolic reflector with feed located in the focal point. The radius of the circularly symmetric aperture is a = D/2.

where E0 is a constant. The electric field in a point P far away from the aperture, can be calculated by using expresion (4.163)

−k e−k0r Z h i h i E~ (~r) = 0 ~e × ~e × E~ (~r ) − ~e × ~e × Z H~ (~r ) ek0(~r0·~er)dS. (4.180) 4πr r n a 0 r n 0 a 0 Sa We have already shown in section 4.2 that the electric field E~ in the far-field region only has two h i h i components: Eθ and Eφ. Therefore, we can express ~er × ~en × E~a(~r0) − ~er × ~en × Z0H~ a(~r0) in the following form:

h i h i M~ = ~er × ~en × E~a − ~er × ~en × Z0H~ a

(4.181) = E0~er × (~ez × ~ex) − ~er × (~ez × ~ey)

= E0(1 + cos θ)(~eφ sin φ − ~eθ cos φ) .

Furthermore, the inner product (~r0 · ~er) can be written in polar coordinates according to

(~r0 · ~er) = r0 sin θ cos(φ − φ0). (4.182) The electric field in the far-field region then takes the following form 2π a −k e−k0r Z Z E~ (~r) = 0 (1 + cos θ)(~e sin φ − ~e cos φ) E ek0r0 sin θ cos(φ−φ0)r dr dφ . (4.183) 4πr φ θ 0 0 0 0 0 0 106 CHAPTER 4. ANTENNA THEORY

Since we have assumed an uniform aperture illumination, E0 is constant and can be put outside the integral. The φ0 integral can be expressed in terms of a Bessel function according to

2π Z k0r0 sin θ cos(φ−φ0) e dφ0 = 2πJ0(k0r0 sin θ), (4.184) 0 where J0 is the Bessel function of the first kind with order zero. The remaining integration w.r.t. r0 can now be expressen in closed-form as:

a 1 Z Z 2 J0(k0r0 sin θ)r0dr0 = a J0(k0aρ sin θ)ρdρ 0 0 (4.185) J (k a sin θ) = a2 1 0 , k0a sin θ where J1 is the Bessel function of the first kind with order one. Inserting this in (4.183) provides

2 −k r a k0E0e 0 J1(k0a sin θ) Eθ = (1 + cos θ) cos φ , 2r k0a sin θ (4.186) 2 −k r −a k0E0e 0 J1(k0a sin θ) Eφ = (1 + cos θ) sin φ . 2r k0a sin θ The radiation pattern is found by using relations (2.8)- (2.10):

2 2 |Eθ(θ, φ)| + |Eφ(θ, φ)| F (θ) = 2 2 |Eθ(0, 0)| + |Eφ(0, 0)| (4.187) J (k a sin θ)2 = (1 + cos θ)2 1 0 . k0a sin θ J (z) 1 Note that we used the relation lim 1 = . The radiation pattern (in [dB]) is shown in Fig. x→0 z 2 4.24 in case of a uniformly-illuminated aperture with a diameter D = 2a = 4λ0. The cross section in the φ = 0-plane is shown in Fig. 4.25. The first sidelobe is found at a level of −17.6 dB w.r.t. 0 the main beam and the 3-dB beam width is θHP ≈ λ0/(2a) = 14.3 . The beam width scales directly with the size of the reflector: the larger D, the smaller the beam width. Note that the first sidelobe of a circular aperture is lower than the first sidelobe of a rectangular apertuur (-17.6 dB versus -13.3 dB).

Up to now, we have assumed a uniform illumination of the aperture: we have we assumed E0 to be constant over the aperture surface Sa. We will now extend our analysis to a more general aperture distribution. For that purpose, we will assume that the aperture distribution on Sa can be expressed in the following form:

E~a(r0) = W (r0)~ex, (4.188)

Z0H~ a(r0) = W (r0)~ey, 4.11. REFLECTOR ANTENNAS 107

Figure 4.24: Three dimensional radiation pattern versus (θ, φ) of a parabolic reflector with uniform apeture illumination with diameter D = 2a = 4λ0. All values are in dB.

Uniform circular aperture, D=2a=4λ 0 0

−5

−10

−15

−20

−25

−30

Normalised radiation pattern [dB] −35

−40

−45

−50 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 4.25: Radiation pattern in the φ = 0-plane θ of a parabolic reflector with uniform apeture illumination with diameter D = 2a = 4λ0. All values are in dB. The 3dB beamwidth in this plane 0 is θHP ≈ λ0/(2a) = 14.3 with aperture distribution (or weighting) function given by

" #p r 2 W (r ) = 1 − 0 . (4.189) 0 a

The specific case p = 0, which provides a uniform illumination, is already discussed in this 108 CHAPTER 4. ANTENNA THEORY section. For other valus of p we have to return to expression (4.183) and replace E0 by the aperture distribution function W (r0). This provides us with the following integral

a 2π" #p Z Z Z r 2 W (r )ek0(r0·~er)dS = 1 − 0 ek0r0 sin θ cos(φ−φ0)dφ r dr . (4.190) 0 a 0 0 0 Sa 0 0

The φ0 integration is done in a similar way as in (4.184). As a result we obtain the following integral along r0: a " #p Z r 2 2π 1 − 0 J (k r sin θ)r dr . (4.191) a 0 0 0 0 0 0

Substituting ρ = r0/a and ua = k0a sin θ results in

1 Z p 2 h 2i 2πa 1 − ρ J0(uaρ)ρdρ. (4.192) 0 This integral can be expressed in terms of a Bessel function by applying the integral of Sonine (see [16], p.373)

π/2 ν+1 Z z µ+1 2ν+1 Jµ+ν+1(z) = ν Jµ(z sin θ) sin θ cos θdθ, (4.193) 2 Γf (ν + 1) 0

Here Γf is the Gamma function [14] for which Γ(p + 1) = p!. Now let µ = 0, ν = p, z = ua and sin θ = ρ. We then find

1 Z p Jp+1(ua) 1 h 2i p+1 = p J0(uaρ) 1 − ρ ρdρ. (4.194) ua 2 Γf (p + 1) 0 As a result, the integral takes the form Z J (u ) k0(r0·~er) 2 p p+1 a W (r0)e dS = 2πa 2 p! p+1 . (4.195) ua Sa The electric field in the far-field region is now easily found by inserting (4.195) in (4.183). Some calculated results are summarized in table 4.1. From this table we can conclude that the uniform illumination (p = 0) provides the highest directivity and antenna gain. However, the first sidelobe level is relative high. By using a weighted aperture distribution (tapering), we can reduce the sidelobe level at the expense of a reduced directivity and corresponding efficiency, which is as low as 44% for p = 2. The requirements for a particular application will determine the optimal shape of the aperture distribution function W (r0).

4.12 Microstrip antennas

Microstrip antennas, also known as patch antennas or printed antennas are resonant antennas which can be realised on printed-circuit boards (PCBs). In order to determine the radiation 4.12. MICROSTRIP ANTENNAS 109

Antenna parameter p = 0 p = 1 p = 2

λ λ λ 3-dB beam width in degrees 29.2 0 36.4 0 42.1 0 a a a λ λ λ Beam width between zeros in degrees 69.9 0 93.4 0 116.3 0 a a a

First sidelobe level [dB] -17.6 -24.6 -30.6

2πa2 2πa2 2πa2 Directivity 0.75 0.56 λ0 λ0 λ0

Table 4.1: Overview of the most relevant antenna parameters as a function of the order p of 2 p the aperture distribution function W = [1 − (r0/a) ] in case of a parabolic reflector antenna of diameter D = 2a. properties of microstrip antennas, we can use the same method as used for aperture antennas in the previous sections. Microstrip antennas can take many forms and shapes. We will limit ourselves to some basic structures. Fig. 4.26 shows the side- and top-view of a rectangular and circular microstrip antenna fed by a coaxial cable. The inner conductor of the coaxial feed is connected to the metal patch and the outer conductor is connected to the ground plane. The dielectric substrate has a thickness h and dielectric constant r. The dielectric losses are indicated by the loss tangent tan δ and will be neglected here. Microstrip antennas can be manufactured with standard PCB etching techniques to realise printed copper structures with accuracies well below 100µm. The patch can also be fed by a microstrip line or by using a resonant slot in the ground plane. In literature, various models have been introduced to determine the radiation properties of mi- crostrip antennas, including analytical models such as the cavity model and transmission line model, and numerical models such as the method of moments (MoM)[17]. In this section we will explore the cavity model in more detail for the case of a circular microstrip antenna as illustrated in Fig. 4.26. We will assume that all metal surfaces (patch and ground plane) are perfect electric conductors (PEC). Furthermore, we will assume that the thickness of the substrate h is very thin as compared to the wavelength, i.e. h  λ0. We will use cylindrical coordinates with (ρ, φ, z) to describe the antenna configuration. With these assumptions, we can introduce the following simplifications:

i. The electric field will be concentrated between the circular patch and the ground plane within a cylinder with radius a and height h. This cylinder is called cavity. The cavity is filled with

a homogeneous diectric material with permittivity r.

ii. Due to the small thickness of the substrate, the fields within the cavity will not depend on the z-coordinate. iii. The electric field has only a component in the z direction, so E~ = Ez~ez. 110 CHAPTER 4. ANTENNA THEORY

z P patch

θ

εr h r

ground plane coax feed

patch a h

φ ρ

y

εr ground plane x

a) Side view and top view of rectangular microstrip antenna b) Circular microstrip antenna

Figure 4.26: Rectangular and circular microstrip antenna fed by a coaxial cable. The metallic patch is printed on a substrate with permittivity r.

iv. The current distribution J~e along the edge ρ = a of the circular patch will be directed parallel to the edge of the patch. This implies that the magnetic field given by H~ = ~ez × J~e, will be perpendicular to the side walls of the cavity, so ~en × H~ = ~0. Note that ~en is the normal on the side walls of the cavity. With these assumptions, our problem is reduced to a cavity structure with the following boundary conditions:

~en × H~ = ~0 on the sidewalls (4.196)

~en × E~ = ~0 on the top and bottom side of the cavity, where ~en is the normal vector on the surface under consideration. Since E~ will only have a z- component, we can express the condition that ~en × H~ = ~0 when ρ = a in terms of a condition for Ez, since −1 H~ = ∇ × E.~ (4.197) ωµ0 As a result, we can express the components of the magnetic field in the following form:

−1 1 ∂Ez Hρ = , ωµ0 ρ ∂φ (4.198) 1 ∂Ez Hφ = . ωµ0 ∂ρ 4.12. MICROSTRIP ANTENNAS 111

Apparantly, on the side walls the following condition holds

∂E z = 0, voor ρ = a. (4.199) ∂ρ

Inside the cavity, we can derive the Helmholtz equation for the electric field E~ = Ez~ez by using Maxwell’s equations for a source-free region (4.5). We now obtain

∇2E~ + k2E~ = ~0, (4.200) √ where k = k0 r represents the wave number in the substrate . The Helmholtz equation in ∂E cylindrical coordinates then takes the following form (with z = 0) ∂z ∂2E 1 ∂E 1 ∂2E z + z + z + k2E = 0. (4.201) ∂ρ2 ρ ∂ρ ρ2 ∂φ2 z The solution of this second-order differential equation is

Ez = E0Jn(kρ) cos(nφ), (4.202) where Jn(kρ) is the Bessel function of the first kind with order n. The other field components are given by

n Hρ = E0Jn(kρ) sin(nφ), ωµ0ρ (4.203) k ∂Jn(kρ) Hφ = E0 cos(nφ). ωµ0 ∂ρ

All other field components are zero, so Eρ = Eφ = Hz = 0. The side wall of the cavity acts as a magnetic wall with ~en × H~ = ~0 or Hφ = 0. This provides us the resonance condition: ∂J (ka) n = 0. (4.204) ∂ρ ∂J (ka) Since n has multiple zeros for a particular values of n, we will find multiple resonance ∂ρ ∂J (ka) frequencies for a given radius a and n. Now let K be the m-th zero of n . Table nm ∂ρ 4.2 summarizes a number of zeros with corresponding values of ka. The resonance frequency corresponding to a particular mode can be determined from the following equation 2πf √ ka =  a = K , (4.205) c r nm Resulting in

Knmc fnm = √ . (4.206) 2πa r The mode with (n, m) = (1, 1) provides the lowest resonance frequency for a particular radius a of the circular patch. Note that the cavity model does not account for fringing fields at the edges of the cavity. These fringing fields will slightly change the resonance frequency. This effect can 112 CHAPTER 4. ANTENNA THEORY

Mode (n, m) zero ka

0, 1 0

1, 1 1.841

2, 1 3.054

0, 2 3.832

3, 1 4.201

4, 1 5.317

1, 2 5.331

5, 1 6.416

∂J (ka) Table 4.2: Zeros of n . ∂ρ

be included in formula (4.206) by introducing an effective radius ae. Closed-form expressions for ae can be found in literature, for example in [18]-[19]. Near resonance it is quite easy to match the input impedance of a microstrip antennas to a transmission line, such as a 50Ω coax cable. The bandwidth of microstrip antennas (defined as the frequency band over which the antenna can be matched to a transmission line with a characteristic impedance Z0) strongly depends on the electrical thickness of the substrate. An accurate estimation of the input impedance of microstrip antennas can be best obtained using numerical methods, such as the Method-of-Moments (MoM), see also chapter 5. Fig. 4.27 shows the realised bandwidth of a coax-fed microstrip antenna versus electrical thickness, according to MoM simulation from [17]. A larger bandwidth can be obtained by using stacked patches [17] or by using a resonant slot feed in the ground plane. The radiation properties of microstrip antennas can be easily determined by using the aperture method according to the Lorentz-Lamor theorem. We will select the surface S as Sp + Sa + Sg, where Sp is the patch, Sa the side wall of the cavity and Sg the ground plane. We can also apply the mirror principle (see section 4.5) in order to account for the ground plane. At the side wall of the cavity we then obtain the following magnetic current distribution

(4.207) J~ms = 2E~ × ~en = 2Ez~eφ voor ρ = a en 0 < z < h. 4.12. MICROSTRIP ANTENNAS 113

Figure 4.27: Bandwidth of a single-layer microstrip antennas versus electrical height of the dielec- tric substrate.

As a result, we find that the field components Eθ and Eφ in the far-field region are given by

n −k r  hak0E0Jn(ka)e 0 Eθ = cos nφ [Jn+1(k0a sin θ) − Jn−1(k0a sin θ)] , 2r (4.208) nhak E J (ka)e−k0r E = 0 0 n cos θ sin nφ [J (k a sin θ) + J (k a sin θ)] , φ 2r n+1 0 n−1 0 where we have assumed that the patch is fed by a transmission line in the φ = 0 plane. For the fundamental mode with (n, m) = (1, 1) we obtain:

−k r hak0E0Jn(ka)e 0 Eθ = cos φ [J2(k0a sin θ) − J0(k0a sin θ)] , 2r (4.209) hak E J (ka)e−k0r E = 0 0 n cos θ sin φ [J (k a sin θ) + J (k a sin θ)] . φ 2r 2 0 0 0 Fig. 4.28 shows the radiation pattern in the φ = 00 plane and the φ = 900 plane of a circular mi- crostrip antenna operating in the (n, m) = (1, 1)-mode and resonating at f = 12 GHz. The radius 0 of the patch is a = 4.6mm and substrate permittivity r = 2.56. The 3-dB beam width is 104 in the φ = 00 plane and 800 in the φ = 900 plane. Due to its omni-directional character, microstrip antennas are very suited in phased-array antennas with wide-scan electronic beamforming. 114 CHAPTER 4. ANTENNA THEORY

Circular Microstrip Antenna, a=4.6mm, f=12GHz, ε =2.56 r 0

−2

−4

−6

−8

−10

−12

Normalised radiation pattern [dB] −14 φ=00 0 −16 φ=90

−18

−20 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 4.28: Radiation pattern of a circular microstrip antenna printed on an electrically thin substrate in the φ = 0 plane and φ = 900-plane. The antenna operates in the fundamental

(n, m) = (1, 1)-mode at f = 12 GHz. The radius of the patch a = 4.6mm and r = 2.56. All values are plotted in dB. Chapter 5

Numerical electromagnetic analysis

In chapter 4 we have determined the far-field radiation characteristics of several types of antennas with a given electric and/or magnetic current distribution. In section 4.4 we have investigated wire antennas, where we assumed a sinusoidal current distribution along the wire. When the antenna is operated close to its resonance, this will be a very good approximation of the real current distribution. However, at other frequencies the current distribution will be different. In general, it is not easy (or not possible) to derive an accurate analytic description of the currents on antenna structures, even not for a fairly simple λ0/2 dipole (wire) antenna. For that purpose, antenna designers use numerical methods to determine the current distribution on the antenna structure. Several numerical approaches are described in literature, including the Finite-Element- Method (FEM), Finite-Difference-Time-Domain (FDTD) and the Method-of-Moments (MoM), see for example in [20]. One of the more common methods for antenna analysis is MoM, which is implemented in the commercial microwave and antenna design tool Momentum which is part of the EDA tool Advanced Design System (ADS) [21]. In this section we will show in a step-by- step approach how MoM can be used to determine the current distribution of a cylindrical wire antenna.

5.1 Moment of moments (MoM)

The method of moments is a general numerical procedure for solving integral equations. One of the first books which describes the MoM for electromagnetic problems is the book of Harrington [22]. In this section we will illustrate how the MoM can be used to determine the current distribution along a simple cylindrical wire antenna. Note that the MoM can also be used to determine magnetic currents. The following step-wise approach is used:

• determine the integral equation of the antenna problem at hand,

• discretize the antenna structure in small segments (also known as sub-domains),

• derive the matrix equation for the (still) unknown current distribution along the antenna,

• calculate the elements of the matrix and solve the matrix equation.

115 116 CHAPTER 5. NUMERICAL ELECTROMAGNETIC ANALYSIS

Now consider the cylindrical wire antenna in free space as illustrated in Fig. 5.1. The antenna is symmetrically oriented along the z-axis with length 2l and diameter 2a. The antenna is fed in the center at z = 0. This is exactly the same structure as discussed in section 4.4. We will

z

2l J

y

2a

x

Figure 5.1: Electrically-thin cylindrical wire antenna of length 2l and diameter 2a. The center of the wire is located at the origin of the coordinate system and is oriented along the z-axis. assume that the diameter of the wire is much smaller than the wavelength, so 2a  λ0. Therefore, we can ignore electric currents at the end surfaces of the cylinder, and as a result, the electric current distribution will only have a component along the z-axis, i.e. J~ = Jz~ez. Note that we have removed the subscript e in J~ in order to simplify the notation in the rest of this section. The antenna configuration of Fig. 5.1 imposes two boundary conditions:

i The tangential field at the perfectly-conducting cylinder with diameter 2a has to be equal to zero.

ii The current at both ends of the cylinder is zero.

In our MoM approach we will only apply the first boundary condition directly. This boundary condition can be describes as

tot ex s (5.1) ~en × E~ (~r) = ~en × (E~ (~r) + E~ (~r)) = ~0, ~rS0,

ex s where the surface S0 is the outer surface of the cylinder and where E~ (~r) and E~ (~r) are the excitation field and the scattered field, respectively. Furthermore, ~en is the normal vector on the metal cylinder. The scattered field is the field due to the induced currents on the wire antenna. The excitation field is the field induced by the voltage source which is connected to the differential port at the center of the antenna. The scattered field can be expressed in terms of the (yet) unknown electric current distribution J~ and the free-space Greens’ function G(~r, ~r0) by 5.1. MOMENT OF MOMENTS (MOM) 117 using expressions (4.30), (4.21) and (4.22) from chapter 4. In case the antenna is not located in free-space, i.e. in case of a microstrip antenna, we can still use the proposed approach. However, the Greens function will be more complicated, see for example in [17] for the case of multi-layer microstrip antennas. For our free-space configuration of Fig. 5.1 we can express equation (5.1) in the following form:

ZZ ex ~en × E~ (~r) = ~en × ωµ0 G(~r, ~r0)J~(~r0)dS0 (5.2)  ZZ  ωµ0 ~ +~en × ∇ 2 ∇ · G(~r, ~r0)J(~r0)dS0 , ~rS0. k0 Integral equation (5.2) can be solved numerically by using the method of moments. The first step is the expansion of the yet unknown current distribution J~ on the wire antenna in expansion functions according to

X J~(x, y, z) = InJ~n(x, y, z), (5.3) n where In are the mode coefficients which we need to determine. The expansion functions J~n(x, y, z) are also called basis functies and can take many forms. When we want to have an exact solution of the surface current distribution J~, we need to use an infinite summation in (5.3). In addition, the set of basis functions need to be complete in the sense that all physical effects are included. In practice we will limit the summation in (5.3) to a maximum number of basis functions of n = Nmax. In this way, we can approximate the exact solution with a limited computational effort. There are many types of basis functions. We can subdivide them into two main categories, that is in global basis functions and local basis functions. Global basis functions are non-zero over the entire domain, in our example over the entire cylindrical wire, whereas local basis functions are only non-zero over a very small part of the entire domain. For that reason, local basis functions are also known as sub-domain basis functies. An example of sub-domain basis functions is shown in Fig. 5.2, where overlapping piece-wise linear (PWL) basis functions are shown that approximate a particular function.

When using Nmax basis functions to describe the yet unknown current distribution on the antenna, provides

N Xmax J~(x, y, z) = InJ~n(x, y, z). (5.4) n

The scattered electric field E~ s(x, y, z) can be written in terms of the current distribution J~(x, y, z)

E~ s(x, y, z) = L{J~(x, y, z)}, (5.5) where L is a linear operator. By combining (5.5) and (5.4) we obtain

Nmax Nmax ~ s X ~ X ~ s E (x, y, z) = InL{Jn(x, y, z)} = InEn(x, y, z). (5.6) n=1 n=1 118 CHAPTER 5. NUMERICAL ELECTROMAGNETIC ANALYSIS

Piece-wise constant (PWC) basis functions Piece-wise linear (PWL) basis functions

z0 z1 z2 z3 z4 z5 z0 z1 z2 z3 z4 z5

Figure 5.2: Piece-wise constant (PWC) and Piece wise linear (PWL) approximation of a function using four PWL basis functions.

Substitution of the expansion (5.6) in equation (5.1) provides

Nmax ! X ~ s ~ ex ~ ~en × InEn(x, y, z) + E (x, y, z) = 0 , (5.7) n=1 on the outer surface S0 of the wire antenna. Let us now introduce the residue R~ according to

Nmax ! ~ X ~ s ~ ex R(x, y, z) = ~en × InEn(x, y, z) + E (x, y, z) . (5.8) n=1 This residue has to be zero over the entire outer surface of the wire antenna. We will relax this requirement by weighting the residue to zero with respect to some weighting functions J~m(x, y, z), such that ZZ (5.9) hR~ ; J~mi = R~(x, y, z) · J~m(x, y, z) dS = 0 , Sm for m = 1, 2..., Nmax, where Sm represents the surface on which the weighting function J~m is nonzero. The set of weighting functions J~m, is also known as a set of test functions. Often the set of test functions is the same as the set of expansion functions in (5.3). This particular choice is called the Galerkin’s method [15]. Inserting (5.8) in (5.9) provideds the following set of Nmax linear equations

Nmax ZZ ZZ X ~ s ~ ~ ex ~ In En(x, y, z) · Jm(x, y, z) dS + E (x, y, z) · Jm(x, y, z) dS = 0, (5.10) n=1 Sm Sm 5.1. MOMENT OF MOMENTS (MOM) 119 for m = 1, 2..., Nmax. We can write this set of linear equations in a more compact form:

Nmax X ex InZmn + Vm = 0, (5.11) n=1 for m = 1, 2..., Nmax. In matrix notation we now obtain: [Z][I] + [V ex] = [0], (5.12) where the elements of the matrix [Z] and the elements of the excitation vector [V ex] are given by

ZZ ~ s ~ Zmn = En(x, y, z) · Jm(x, y, z) dS, Sm (5.13) ZZ ex ~ ex ~ Vm = E (x, y, z) · Jm(x, y, z) dS, Sm

The matrix [Z] includes Nmax × Nmax elements, [I] is a vector with Nmax unknown mode coeffi- ex cients and [V ] is the excitation vector with Nmax elements. The matrix equation (5.12) can be solved rather easily by using standard numerical linear-algebra routines such as those available in MATLAB. In practise it will turn out that the largest computational effort is claimed for the cal- culation of the elements of the matrix [Z]. After matrix equation (5.12) has been solved, we obtain a solution for the mode coefficients and we can find an approximation of the current distribution J~ on the antenna surface by inserting the mode coefficients in expression (5.4). Once the current distribution is known, we can calculated the far-field pattern and other antenna characteristics by using the concepts from chapters 2 and 4. The elements of the matrix [Z] can be found by using the expression for E~ s as applied in (5.2), which results in:

ZZ  ZZ Zmn = −ωµ0 G(~r, ~r0)J~n(x0, y0, z0)dS0 Sm Sn (5.14)  ZZ  ωµ0 ~ ~ −∇ 2 ∇ · G(~r, ~r0)Jn(x0, y0, z0)dS0 · Jm(x, y, z) dS, k0 Sn In general it will not be possible to find an analytic expression for the elements of [Z]. We will need to use numerical methods. Note that, depending on the particular antenna structure under investigation, several numerical problems (e.g. singularities) might occur while evaluating (5.14). More on this can be found in [20] and other literature. We will now return to our example, the cylindrical wire antenna as illustrated in Fig. 5.1. Since we have assumed that the diameter 2a  λ0, the electric current distribution will only have a component along the z-axis, i.e. J~ = Jz~ez. Therefore, the expansion and test functions will only have a component in the z direction, so J~m = Jzm~ez. We can now write an element of the matrix [Z] in the following form ! Z 1 ∂2 Z Zmn = −ωµ0 1 + 2 2 G(z, z0)Jzn(z0)dz0Jzm(z) dz k0 ∂z m n (5.15) ! Z Z 1 ∂2 = −ωµ 1 + G(z, z )J (z )dz J (z) dz 0 k2 ∂z2 0 zn 0 0 zm m n 0 120 CHAPTER 5. NUMERICAL ELECTROMAGNETIC ANALYSIS where the Green’s function G(z, z0) is given by

e−k0Ra G(z, z0) = , (5.16) 4πRa with q 2 2 Ra = a + (z − z0) . (5.17)

Note that Ra is nonzero for z = z0. This is the result of our choice (in fact approximation) to define the expansion function on the outer cylindrical surface (ρ = a) of the wire antenna, whereas the test functions are defined along the axis of the wire antenna (ρ = 0). In this way, we avoid a singularity of the Green’s function G(z, z0) at z = z0. The integral equation for the wire antenna that leads towards expression (5.15) for Zmn is known as the Pocklington integral equation. Next to this equation, we can also derive an alternative integral equation, known as the Hall´enintegral equation, see also [22],[20]. In our example, we will choose piece-wise constant (PWC) expansion- and test functions. The m-th basis function of this set is defined as     1, zm−1 < z < zm+1, Jzm(z) = (5.18)    0, elsewhere,

where the index m = 1, 2, .., Nmax. For the wire antenna of Fig. 5.1, we get z0 = −l en zNmax+1 = l. Note that the PWC basis functions do not explicitly satisfy the physical requirement that the current at both ends of the wire should be equal to zero. This requirement can be satisfied when using PWL basis functions (see Fig. 5.2). For that reason, PWL basis functions will provide a more accurate result for a given maximum of sub-domains. By substituting (5.18) for PWC functions in the expression for Zmn (5.15), we obtain

zm+1 zn+1 ! Z Z 1 ∂2 e−k0Ra Zmn = −ωµ0 1 + 2 2 dz0 dz (5.19) k0 ∂z 4πRa zm−1 zn−1

The double derivative w.r.t. z in (5.19) can easily be obtained analytically, see [23], [19]. The resulting double integral then needs to be determined numerically, for example by using standard integration routines available in MATLAB. The last step in the calculation of the mode coefficients with (5.12) is the calculation of the excitation vector [V ex]. The excitation field in expression (5.13) can be an incident plane wave or a locally generated field. In this example, we will assume that a voltage generator is connected to the input port of the wire antenna, located at the center of the wire. The voltage generator can be approximated by a so-called delta-gap generator, which generates locally a very strong electric field directed along the z-axis. This excitation field can be expressed as follows:

ex E~ = Vgδ(z − zg)~ez, (5.20) 5.1. MOMENT OF MOMENTS (MOM) 121 where Vg represent the generator voltage and where zg is the position of the delta-gap generator. When we connect the delta-gap generator at the center of mode m = i, with zg = zi, the excitation vector [V ex] takes the following form:      V ex   0   1                 V ex   0   2                     ......   ......  ex     [V ] =   =   . (5.21)      ex     Vi   Vg                   ......   ......               V ex   0  Nmax

As a result, only element i of the vector [V ex] is nonzero. We have now determined all elements of [Z] and [V ex] in matrix equation (5.12). By solving this equation (numerically), we can determine the unknown mode coefficients and related current distribution along the wire antenna. The far- field pattern is then determined by using the expressions introduced in section 4.2. The input impedance is easily found as the relation between the input voltage and input current:

Vg Zin = , (5.22) Ii where Ii is the mode coefficient of mode i. 122 CHAPTER 5. NUMERICAL ELECTROMAGNETIC ANALYSIS Chapter 6

Phased Arrays and Smart Antennas

Phased-array antennas with electronic beamsteering capabilities have been developed in the period between 1970-1990 mainly for military radar applications to replace mechanically rotating antenna systems. Reflector antennas have a high antenna gain, but have the disadvantage that the main lobe of the antenna has to be steered in the desired direction by means of a highly accurate mechanical steering mechanism. This means that simultaneous communication with several points in space is not possible. Small antennas, like a dipole or microstrip antenna have a close to omni- directional radiation pattern, but have a very low antenna gain. Therefore, small antennas cannot be used for large-distance communication or sensing. By combining a large number of small antennas, we can create an array of antennas. When each of these individual antenna elements is provided with an adjustable amplitude and phase, we obtain a phased-array antenna. In this way, a phased-array antenna can to communicate with several targets which may be anywhere in space, simultaneously and continuously, because the main beam of the antenna can be directed electronically into a certain direction. Another advantage of phased-array antennas is the fact that they are usually relatively flat. Examples of modern phased array systems are shown in Fig. 6.1 and Fig. 6.2 used in radar and radio astronomy, respectively. Phased arrays are also very suited for future millimeter-wave 5G and beyond 5G wireless communication systems, where a high antenna gain is required in combination with electronic beamsteering. In this chapter, we will first investigate the general properties of phased arrays, starting with linear arrays of isotropic radiators. Then, we will extend the theory to two-dimensional arrays. As a next step, we will introduce real antenna radiators, like dipoles and microstrip antennas. We will show that mutual coupling between the elements can limit the performance of the array. Finally, we will investigate the effect of errors and present a method to calibrate the array to compensate for errors. In this chapter we will again assume to have time-harmonic signals and waves.

6.1 Linear phased arrays of isotropic antennas

6.1.1 Array factor

We will investigate the basic principle of operation of linear arrays by considering the array in receive mode, as illustrated in Fig. 6.3. In principe, due to reciprocity, the behaviour of the

123 124 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.1: Example of a phased-array radar operating at X-band frequencies, 6.33

Figure 6.2: Example of a large phased-array antenna used in radio astronomy, [13] array antenna in transmit mode is exactly the same. However, in practical active phased-array systems, the functionality of the transmit electronics may differ from the receive electronics. The linear array of Fig. 6.3 consists of K identical antenna elements with k = 1....K, separated by an element spacing of dx. Element k = 1 is located in the origin of the coordinate system at (x, y, z) = (0, 0, 0). In this section, we will assume that all antenna elements are isotropic (omni-directional) radiators. Although isotropic radiators cannot exist in reality, they allows us to investigate the basic array properties. Each antenna element k includes a circuit which can perform a complex weighting of the received signal sk, by multiplication with the complex −ψk coefficient ak = |ak|e . This complex coefficient consists of an amplitude |ak| (Eng: taper) and a phase shift −ψk. The set of complex coefficients are also often called array illumination. In general the complex weighting will be realized by means of an electronic circuit and can be done 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 125 both in the analog and digital domain. Finally, all signals are summed in the summing network.

z W 0 A VE F RO Antenna NT element 0 in s d x

dx dx s1 s2 sK-1 sK

-j 1 -j 2 -j K-1 -j K a1 e a2 e aK-1 e aK e

SUMMINGNETWORK

S

Figure 6.3: Linear phased-array antenna consisting of K identical antenna elements, separated by a distance dx. The signals received by each of the antenna elements are multiplied by a complex coefficient with amplitude |ak| and phase −ψk.

Now let us investigate the receive properties of a linear array of isotropic antennas in more detail. We will assume that a time-harmonic electromagnetic plane wave is incident on the array from an angle θ0 with respect to the z-axis. This plane wave can be assumed to be generated by another (transmit) antenna located far away from the array. The received signal by array element k can now be written as:

k0(k−1)dx sin θ0 sk = e , (6.1) 2π where we have normalized the received signal s1 of element 1 to be equal to 1. In addition, k0 = λ0 is the free-space wavenumber. The received signals at each array element can be physically interpreted as the voltage along the output port of each antenna element or can represent the (complex) amplitude of the guided wave along the transmission line which is connected to the antenna element, for example in case of a waveguide or a microstrip transmission line. Since the incident plane wave illuminates the array with an angle θ0, a phase difference will occur between the signals received by each of the array elements. This phase difference could also be interpreted as a time delay, since element k will receive the signal a fraction τ earlier as compared to array element (k − 1), where τ is given by:

d sin θ τ = x 0 , (6.2) c where c = 3.108 is the speed of light in free-space. By using the superposition concept of electro- 126 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS magnetic waves we can write the total received signal of the array as:

K X [k0(k−1)dx sin θ0−ψk] S(θ0) = |ak|e . (6.3) k=1

The function S(θ0) is called the arrayfactor. The array factor is the response of the array of Fig. 6.3 on an incident plane wave that hits the array surface under the angle θ0, expressed in the −ψ coefficients ak = |ak|e k with k = 1...K − 1. The array factor is a periodic function of sin θ0 with a period of λ0/dx. A similar analysis could be done for an array in transmit mode. Besides some constants and the factor e−k0r/r, we would obtain a similar expression as (6.3) for the electromagnetic field far away from the array (far field).

The maximum of the array factor occurs when the applied phase shift −ψk by the electronics connected to each of the receiving antenna elements exactly compensates the phase of each of the received signals. This occurs when:

ψk = k0(k − 1)dx sin θ0. (6.4)

By changing the set of applied phases ψk, we can manipulate the direction θ0 for which the array is maximized. In other words, by changing the set of phases ψk, we can electronically scan the beam of the array. Because of this, this type of antenna is called a phased array.

Let us now assume that we use a set of excitation coefficients ak of which the phase ψk is given by (6.4). The resulting array will have a maximum sensitivity when incident plane waves hit the array with an angle of θ = θ0 w.r.t. to the z-axis. Now let us introduce the variables u en u0 according to:

u = sin θ (6.5)

u0 = sin θ0

If we now assume that a plane wave is incident on the array under an angle θ, we can re-write the array factor (6.3) by means of the following expression

K X [k0(k−1)dx(sin θ−sin θ0)] S(u) = |ak|e k=1 (6.6) K X [k0(k−1)dx(u−u0)] = |ak|e . k=1

6.1.2 Uniform amplitude tapering

In case of an uniform amplitude weighting (tapering), where |ak| = 1 for k = 1...K, we can express the summation of (6.6) in terms of an analytical expression. By introducing β = k0dx(u − u0), we can re-write the array factor:

K S(β) = X e(k−1)β. (6.7) k=1 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 127

By multiplying both left- and right-side of (6.7) with eβ we obtain:

K K S(β)eβ = eβ X e(k−1)β = X ekβ. (6.8) k=1 k=1 By subtracting the relation (6.7) from (6.8), we get:   S(β) eβ − 1 = eKβ − 1. (6.9)

In other words: " # eKβ − 1 S(β) = eβ − 1 " # eKβ/2 − e−Kβ/2 = e(K−1)β/2 (6.10) eβ/2 − e−β/2 sin(Kβ/2) = e(K−1)β/2 . sin(β/2) Expressed in u coordinates, we then finally find the following expression for the array factor of an array with uniform tapering:

sin(Kk d (u − u )/2) S(u) = e(K−1)k0dx(u−u0)/2 0 x 0 . (6.11) sin(k0dx(u − u0)/2) The antenna radiation pattern can by found by normalizing the received power w.r.t. its maximum value, according to the definition introduced in 2. When we apply this to our array factor, we obtain the following expression for the normalized (power) radiation pattern of the linear array:

2  2 |S(u)| sin(Kk0dx(u − u0)/2) (6.12) 2 = . |S(u)|max K sin(k0dx(u − u0)/2)

For small values of the term k0dx(u − u0)/2, this reduces to

2  2 |S(u)| sin(Kk0dx(u − u0)/2) (6.13) 2 ≈ . |S(u)|max Kk0dx(u − u0)/2 Expression (6.13) is exactly equal to the radiation pattern of an uniformly illuminated continuous line source of length L = Kdx. This can easilty be verified by comparing (6.13) with expression (4.175), which resembles the radiation pattern of an uniform aperture. As an example, we will investigate in more detail a linear array with uniform tapering consisting of

K = 16 isotropic antenna elements, separated by a distance dx = λ0/2. The normalized radiation pattern can be found by using equation(6.12). Fig. 6.4 shows the radiation pattern in case the 0 main beam of the array is directed towards the angle θ0 = 0 . Fig. 6.5 shows the radiation 0 pattern when the main beam is scanned towards θ0 = 40 . The first sidelobe of the radiation pattern is located at a level of -13.2 dB below the main lobe level. The height of the first sidelobe is directly related to our choice of an uniform tapering. It can be shown that the 3dB beamwidth 0.8858λ0 of a linear array with uniform tapering for large values of K equals: θHP ≈ [24]. With Kdxcosθ0 0 0 K = 16, dx = λ0/2 and θ0 = 0 , results in θHP ≈ 6.3 . The exact 3dB beamwidth in Fig. 6.4 is 0 θHP = 6.2 . 128 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Array pattern of 16−element uniform array with λ /2 spacing 0 0

−5

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−40 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 6.4: Normalized radiation patter according to (6.12) of a uniformly illuminated linear array of 16 isotropic antennas. The main beam is located at broadside (θ = 00). The element spacing is dx = λ0/2.

Array pattern of 16−element uniform array with λ /2 spacing 0 0

−5

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−40 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 6.5: Normalized radiation pattern according to (6.12) of a uniformly illuminated linear array of 16 isotropic antennas. The main beam is located at an angle of θ = 400. The element spacing is dx = λ0/2.

6.1.3 DFT/FFT relation When we investigate expression (6.6) more carefully, we can recognize a Discrete Fourier Trans- formatie (DFT). The DFT can be written in the following general form:

K X 2π(k−1)(n−1) F (u) = f(k)e K 1 ≤ n ≤ K. (6.14) k=1 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 129

We can rewrite the array factor (6.6) in a similar form as (6.14) by choosing the sample points in the u space according to:

λ0(n − 1) u − u0 = . (6.15) dxK For the specific case that the number of array elements K is a power of 2, the array factor simplifies in to a Fast Fourier Transform (FFT). By using FFT processing, we can efficiently compute the radiation pattern of very large phased arrays. FFT processing is also often used in the digital back-end of phased array antennas with digital beamforming.

6.1.4 Grating lobes

Previously, we have already shown that the main beam of a linear array can be scanned to any angle θ0 by applying a linear phase distribution along the array according to (6.4). The array factor as defined in (6.6) is a periodical function of the variable u−u0. In other words: besides the main beam, the array factor can have more maxima in the visible space, the so-called grating lobes. The existence and exact location of the grating lobes w.r.t. the main beam depends on the element spacing dx. In most practical systems, dx will be chosen in such as way as to avoid the existence of grating lobes in the visible space, where the visible space is defined by −1 ≤ u = sin θ ≤ 1.

A grating lobe will appear in the direction um when the argument of the exponent in expression (6.6) can be written as an integer number of 2π

k0(k − 1)dx(um − u0) = 2πm, (6.16) 2π where m is an integer number and where k0 = . For a given scan angle of the main beam λ0 0 u0, a grating lobe will appear in the visible space when |um = 1| of θm = ±90 . The relation max between the element spacing and the maximum scan angle θ0 of the main beam of the linear array without the appearance of a grating lobe is given by:

dx 1 ≤ max . (6.17) λ0 1 + | sin θ0 |

max 0 A lot of applications require scanning over the entire visible space. By substituting θ0 = 90 in expression(6.17), we can observe that grating lobes are avoided when the element spacing dx ≤ λ0/2. This corresponds to the well-known Nyquist sampling criterium used in analog-to- digital convertors (ADC).

6.1.5 Amplitude tapering and pattern synthesis

In section 6.1.2 we have shown that a linear array with uniform amplitude tapering, i.e. |ak| = 1 for all k, generates quite high sidelobes in the radiation pattern, corresponding to the behavior of the sinc function. This results in a first sidelobe level of -13.2 dB w.r.t. the main beam power level. We can reduce the sidelobe level of phased arrays considerably by using an amplitude weighting along the array elements. The weighting function or amplitude taper needs to be chosen in such a way that contribution to the total array factor of the edge elements is reduced. In other words, the 130 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS illumination of the edges is reduced. One of the well-known taper functions is the cosine-power taper given by:   m πxk Kdx |ak| = h + (1 − h) cos , |xk| ≤ , (6.18) Kdx 2 in case of an array along the x-axis, where xk represent the x-coordinate of array element k. The parameter h provides us to use an offset (or Pedestal). When h = 1, we again obtain an uniform taper. Commonly used taper are the cosine taper (m = 1) and the cosinus-square taper (m = 2). The corresponding normalized radiation patterns are shown in Fig. 6.6 and Fig. 6.7 in case of an array with K = 16 elements and element spacing dx = λ0/2. The cosine taper results in

Array pattern of 16−element array with cosine (m=1) taper and λ /2 spacing 0 0 Cosine taper Uniform taper −5

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−50 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 6.6: Normalized radiation pattern of a linear array with a cosine taper (m = 1) with h = 0 and K = 16 array elements. The highest sidelobe is found at a level of -23 dB. The main beam is 0 directed towards broadside θ = 0 . The element distance dx = λ0/2. The corresponding radiation pattern of an array with uniform tapering is also shown. a peak sidelobe level of approx. -23 dB and the cosine-square taper results in a peak sidelobe level of -32 dB. This is a significant improvement as compared to an uniform taper. Other well known taper function are the Taylor taper and the Dolph-Tchebycheff taper [24]. The Taylor tapering results in a monotonic decrease of the sidelobe level away from the main beam, where as the Dolph-Tchebycheff taper creates a constant sidelobe level. An example of the taylor taper is shown in Fig. 6.8 for a 16-element linear array with a spacing of dx = λ0/2. The Taylor taper provides low sidelobes with a peak sidelobe level of 30 dB in this case. As a side effect, the beam width of the main beam increases, and as a result, the directivity will be lower as compared to an array with uniform tapering, see also next section. One of the main benefits of phased arrays is that the pattern (main beam and sidelobes) can be controlled by using specific sets of the complex weight of each of the array elements. Several pattern synthesis methods are described in literature that control the main beam and/or sidelobe level. An overview of these methods can be found in [25]. The synthesis method introduced by 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 131

Array pattern of 16−element array with cosine−squared (m=2) taper and λ /2 spacing 0 0 Cosine taper Uniform taper

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−50 Normalised array pattern [dB]

−60

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−80 −80 −60 −40 −20 0 20 40 60 80 θ [deg]

Figure 6.7: Normalized radiation pattern of a linear array with a cosine-square taper (m = 2) with h = 0 and K = 16 array elements. The highest sidelobe is found at a level of -32 dB. The main 0 beam is directed towards broadside θ = 0 . The element distance dx = λ0/2. The corresponding radiation pattern of an array with uniform tapering is also shown.

Figure 6.8: Normalized radiation pattern of a 16-element linear array with a Taylor taper with 30 dB peak sidelobe level. The main beam is directed towards broadside θ = 00. The element distance dx = λ0/2. The corresponding radiation pattern of an array with uniform tapering is also shown.

Schelkunov [26] provides some physical back-ground in sidelobe-control of pencil-beam patterns. The array factor of a linear array (6.6) can also be written as a polynomial of a new variable z, resulting in the so-called Schelkunov’s polynomial:

2 K−1 (6.19) S(z) = a1 + a2z + a3z + .... + aK a , 132 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

k0dx sin θ −k0dx sin θ0 where z = e and ak = |ak|e . Note that |z| = 1, so located on a unit-circle in the complex z plane. The Schelkunov polynomial (6.19) can also be written in terms of a product of K − 1 factors:

(6.20) S(z) = aK (z − z1)(z − z2).....(z − zK−1), where z1, z2, .... zK−1 represent the zeros of the polynomial. A zero in the polynomial corresponds with a null-location in the radiation pattern of the linear array. As an example, we will consider a linear array with K = 8 elements with spacing of d = λ0/4 and main beam directed towards broadside (θ0 = 0). Let us first consider a uniform amplitude tapering. Fig. 6.9 shows the locations of the zeros in the Schelkunov polynomial plotted in the complex z-plane, together with the radiation pattern. The null at θ = −300 in Fig. 6.9 corresponds with the following location in the complex z-plane:

2π λ k d sin θ  0 sin θ z = e 0 x = e λ0 4 (6.21)  π sin θ  π = e 2 = e 4

Figure 6.9: Zeros of the Schelkunov polynomial plotted in the complex z-plane and relation to the nulls in the radiation pattern. Linear array with uniform illumination (|ak| = 1) K = 8 elements with spacing of d = λ0/4 and main beam directed towards broadside (θ0 = 0).

The separation between the nulls of an uniformly illuminated array in Fig. 6.9 is equidistant. If we want to create low sidelobes, we should try put the nulls closer together and move the first nulls away from the main-lobe. This is done when using a Taylor of Dolph-Chebychev tapering. This is illustrated in Fig. 6.10, where the null location of a Taylor taper with -20 dB peak sidelobe is compared to the uniformly illuminated array. The corresponding radiation pattern and taper function along the array is provided in Fig. 6.11. By comparing the location of the nulls with Fig. 6.11, it can be observed that the nulls have indeed shifted away from the main beam and are now located closer together. In addition, the beam width of the main beam has broadened, which will result in a reduction of the directivity as we will see in the next section. 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 133

Figure 6.10: Zeros of the Schelkunov polynomial plotted in the complex z-plane for a uniform and

Taylor taper. Linear array with K = 8 elements with spacing of d = λ0/4 and main beam directed towards broadside (θ0 = 0).

Figure 6.11: Radiation pattern (left) and amplitude taper along the array (right) of a linear array with K = 8 elements with spacing of d = λ0/4 and main beam directed towards broadside (θ0 = 0).

6.1.6 Directivity and taper efficiency The directivity of an array directly determines the range of a wireless link or radar system as stated by the radio- and radar equation (see chapter 2). The directivity of a linear array of isotropic radiators can be determined by substituting the array factor (6.6) in to equation (2.11).

In this way, we can obtain the directivity in the direction θ0:

P (θ0) D(θ0) = Pt/4π 2 4π|S(θ0)| (6.22) = Z , |S(θ)|2dΩ

V∞ where the integration is done over the entire visible space. The integral can be solved most easily by using a new angle-coordinate ξ, where ξ is defined as the opening angle with the positive 134 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS y-axis in the y − z-plane. The differential solid angle is now given by dΩ = cos θdθdξ. In case of a linear array located along the x-axis, the array factor will not depend on the ξ coordinate. Let 0 us assume that the main beam of the array is directed towards broadside, so θ0 = 0 . By using u = sin θ and du = cos θdθ, we can rewrite (6.22) in the following form:

K !2 X 2 |ak| D = k=1 1 Z |S(u)|2du −1 K !2 X 2 |ak| = k=1 1 Z K K X X k0dxu(k−l) |ak||al|e du −1 k=1 l=1 (6.23) K !2 X 2 |ak| = k=1 1 K K Z X X k0dxu(k−l) |ak||al| e du k=1 l=1 −1 K !2 X |ak| = k=1 . K K X X |ak||al|sinc[k0dx(k − l)] k=1 l=1

When k0dx = mπ (which implies that dx = mλ0/2), we find that sinc[k0dx(k − l)] can only be non zero when k = l. Therefore, we directivity can be expressed as:

K !2 X |ak| D = k=1 . (6.24) K X 2 |ak| k=1

As a result, we can observe that the directivity of an uniformly illuminated linear array with element spacing dx = mλ0/2 is equal to D = K. Consider for example, an array with K = 1000 (isotropic) elements. The corresponding directivity is in this case D = 10 log10(1000) = 30 dB. When we apply an amplitude tapering with |ak| ≤ 1, the directivity will be reduced as compare to the uniform case according to:

(6.25) D = Kηtap, 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 135 where the taper efficiency ηtap is given by

K !2 X |ak| η = k=1 . (6.26) tap K X 2 K |ak| k=1 An example is shown in Fig. 6.12 for a linear array with K = 64 elements and a spacing of dx = λ0/2. The figure shows the radiation pattern in case of a Taylor taper with 40 dB peak sidelobe as compared to the uniform case. The Taylor taper provides low sidelobes, but also a directivity reduction of 1.2 dB (ηapp = 0.76). In other words, the effective antenna aperture has reduced with 24% as compared to the uniformly illuminated array.

Figure 6.12: Effect of Taylor amplitude tapering on the radiation pattern of a linear array with

K = 64 elements with element spacing of d = λ0/4 and main beam directed towards broadside (θ0 = 0). The taper efficiency ηapp = 0.76.

6.1.7 Beam broadening due to beam scanning

The beam width of an antenna has been defined in chapter 2. In this section, we will again consider an uniformly illuminated linear array of isotropic radiators. The array factor is given by expression (6.12). From this expression, we can observe that the array factor depends on the coordinate u−u0, where u0 is a fixed value corresponding to the direction of the main beam (scan angle). Therefore, we can conclude that the array factor does not depend on the absolute value of u. As a result, the 3dB beam width ((i.e. the u value for which the amplitude of the array factor √ is equal to 1/ 2 w.r.t. the maximum value) will also not depend on u. To conclude, the beam width will be constant in the u-domain. However, when we express the array factor is terms of θ, the situation is different since the relation between u en θ is given by:

u = sin θ, (6.27) 136 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS waarmee du dθ = . (6.28) cos θ Een kleine verandering ∆θ rondom θ0 is blijkbaar te schrijven als ∆u ∆θ ≈ . (6.29) cos θ0 Assume that ∆u indicates the difference in u corresponding to the 3dB beamwidth in the expres- sion of the array factor (6.12). The corresponding beam width expressed in θ will then have a 1 dependence. As a result, we can write the 3dB beamwidth in the following form: cos θ0

θ0 θHP = . (6.30) cos θ0 As expected, the beam of the array becomes broader when the main beam is scanned towards the 0 horizon (θ0 = ±90 ).

6.1.8 Phase shifters (PHS) versus true time delay (TTD) Traditionally phased-arrays use phase shifters (PHS) to steer the beam of the antenna. As an alternative true-time delays (TTD) can be used. In order to understand the differences and whether PHS or TTD should be used in an application, let us investigate the array factor in more detail. The general form of the array factor is given by (6.6):

K K  2πf0  X [k0(k−1)dx(u−u0)] X  (k−1)dxu−Ψk S(u) = |ak|e = |ak|e c , (6.31) k=1 k=1 where f0 is the frequency of operation, c is the speed of light and Ψk is the applied phase shift to array element k at the frequency f0. In a PHS unit, the realized phase shift is constant versus frequency. In a TTD unit, the phase shift is realized with transmission-lines of a specific length. As a consequence, the realized phase will depend linearly on frequency. An example of a 4-element beamformer using 2-bit TDU units is shown in Fig. 6.13. When we look carefully at equation (6.31, we can observe that the required phase shift Ψk should have a linear dependency 2πf0 with frequency, since the term c (k − 1)dxu also shows a linear behavior versus frequency. As a consequence, PHS units can only be applied in phased-array systems with a relative small operational frequency bandwidth. At least when a fixed scan angle is required. The effect of so-called beam squinting is illustrated in Fig. 6.14, where the radiation pattern of a 64-element linear array is shown at 10, 12 and 14 GHz. The centre frequency of the array is f0 = 12 GHz at which the optimal phase shift were chosen to scan the array towards 400. The main beam is clearly squinting to another direction when the frequency of operation is changed according to:

f u = 0 , f0 u (6.32) f u = u 0 , 0 f where we assumed that f0 is the design frequency of the array (in this case 12 GHz) and f is the frequency of interest. Note that beam squinting can also be usefull: it is a way to realize beam steering without the need of adjustable phase-shifters. 6.1. LINEAR PHASED ARRAYS OF ISOTROPIC ANTENNAS 137

Figure 6.13: 4-element beamformer using TDU units including 4 dipole-like antenna elements. The switches in the TDU units are realized with MEMS technology [27]

Figure 6.14: Beam squinting in a linear array of 64 elements. Phase-shifters are used optimized at the centre frequency f0 = 12 GHz.By using TDU units, the main beam would remain fixed at 0 θ0 = 40 over the entire frequency range. 138 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

6.2 Planar phased arrays of isotropic radiators

The theory of linear arrays can be easily extended to planar arrays. Fig. 6.15 shows the geometry of a two-dimensional planar array, in which the elements are placed on a rectangular grid. Let us again first assume that all array elements are isotropic radiators. The element distance along the x-axis and y-axis are given by dx and dy, respectively. j denotes the element index according to j = (l − 1)K + k. The position of the j-th element in the array is denoted by ~rj according to:

~rj = xj~ex + yj~ey (6.33)

= (k − 1)dx~ex + (l − 1)dy~ey.

We will again investigate the array in receive-mode first. Now let us assume that a plane wave illuminates the array from the direction (θ, φ), denoted by the unit vector ~er. The array factor can now be obtained by summing all received contributions from the array elements according to:

K L X X [k0(~rj ·~er)−ψj ] S(θ, φ) = |aj|e , (6.34) k=1 l=1 where

−ψj aj = |aj|e , j = (l − 1)K + k, (6.35) represent the excitation coefficient of element j with phase ψj and amplitude |aj|. The inner product (~rj · ~er) can be found by

~rj · ~er = [(k − 1)dx~ex + (l − 1)dy~ey] · [sin θ cos φ~ex + sin θ sin φ~ey + cos θ~ez] (6.36)

= (k − 1)dxu + (l − 1)dyv. with u and v according to

u = sin θ cos φ, (6.37) v = sin θ sin φ.

As a result, we can rewrite the array factor (6.34) in terms of (u, v)-coordinates according to

K L X X [k0(k−1)dxu+k0(l−1)dyv−ψj ] S(u, v) = |aj|e . (6.38) k=1 l=1

When we want to direct the main beam of the array towards (θ0, φ0), the phase shift ψj at each array element should be chosen as:

ψj = k0(k − 1)dxu0 + k0(l − 1)dyv0, (6.39) 6.2. PLANAR PHASED ARRAYS OF ISOTROPIC RADIATORS 139

z

y

dy

Arra y el eme nt j

s x j

dx -j j aj e

Figure 6.15: Planar phased-array antenna consisting of K × L elements separated by a distance dx along the x-axis and dy along the y-axis. The element index is j = (l − 1)K + k. The signals received by each antenna element is fed into an electronic receiver that takes care of the amplitude weighing |aj| and phase shifting −ψj of the received signal.

with u0 = sin θ0 cos φ0 and v = sin θ0 sin φ0. Substituting (6.39) in (6.38), results in the expression for the array factor of a planar array with the main beam directed towards (u0, v0): K L X X k0[(k−1)dx(u−u0)+(l−1)dy(v−v0)] S(u, v) = |aj|e . (6.40) k=1 l=1

Now consider an uniformly illuminated array with |aj| = 1, for all j. The array factor can now be expressed in terms of a product of two array factors corresponding to two linear arrays with uniform illumination:

K ! L ! X X S(u, v) = ek0(k−1)dx(u−u0) ek0(l−1)dy(v−v0) k=1 l=1

= e(K−1)k0dx(u−u0)/2e(L−1)k0dy(v−v0)/2 (6.41) " # sin(Kk d (u − u )/2) sin(Lk d (v − v )/2) × 0 x 0 0 y 0 . sin(k0dx(u − u0)/2) sin(k0dy(v − v0)/2) Fig. 6.16 shows the normalized radiation pattern of a planar array consisting of 64 isotropic antennas (K = L = 8). The element spacing is equal to λ0/2 in both directions. The main 0 0 beam is scanned towards (θ0 = 40 , φ0 = 0 ). We can clearly observe the beam broadening of the main beam and of the first sidelobes. The peak value of the first sidelobe is again -13.2 dB, corresponding to the sinc function in expression (6.41). Lower sidelobe levels can be obtained by applying amplitude tapering, in a similar way as done for linear arrays. 140 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.16: Normalized radiation pattern of an uniformly illuminated planar array of isotropic 0 antennas with 8×8 = 64 elements. The main beam of the array is directed towards (θ0 = 40 , φ0 = 0 0 ). The element spacing is dx = dy = λ0/2.

In our analysis of linear arrays, we have already observed that multiple maxima (grating lobes) can occur in the array factor. The happens when the distance between the array elements is too large. Grating lobes in a planar array occur when the argument in the exponent of (6.40) is a multiple of 2π. As a result, a grating lobes will appear in the direction(um, vm) when the following condition is met:

k0 [(k − 1)dx(um − u0) + (l − 1)dy(vm − v0)] = 2πm, (6.42) for integer values of m, where m = 0 corresponds to the main beam. In case of a rectangular array grid, the grating lobes will occur along a rectangular grid in the (u, v)-plane given by:

i u − u = i = 0, ±1, ±2, .... 0 d /λ x 0 (6.43) j v − v0 = j = 0, ±1, ±2, .... dy/λ0 The rectangular grid of grating lobes in the (u, v)-vlak is illustrated in Fig. 6.17 in case of an dx dx array with = 0.5 and = 1 with the main beam directed towards broadside u0 = v0 = 0. We λ0 λ0 can clearly see that all grating lobes appear outside the unit circle. As a result, no grating lobes will appear in the visible space. In case of scanning, the whole pattern will shift according to the λ arrows as indicated in Fig. 6.17. In case 2, the element spacing is larger than 0 . As a result, 2 grating lobes will appear in the visible space. In most applications grating lobes will deteriorate the performance of the antenna system.

6.3 Arrays of non-ideal antenna elements

Up to now, we have assumed that the individual array element are ideal isotropic radiators with omni-directional radiation characteristics. As already discussed before, isotropic radiators can 6.3. ARRAYS OF NON-IDEAL ANTENNA ELEMENTS 141

Figure 6.17: Grating lobe pattern in the (u, v)-plane of a planar array placed on a rectangular grid d d d d with: case 1 x = y = 0.5 and case 2 x = y = 1. The blue dot show the main beam scanned λ0 λ0 λ0 λ0 towards broadside (u0 = v0 = 0). The entire pattern will shift according to the arrows in case of scanning. physically not exist. Therefore, we need to introduce physical antenna structures in our array theory. However, most of the array features that we have already discussed in this chapter will not change significantly by using other antenna types. Several types of antennas can be used as a building block in a phased array system, such as dipole antennas (usually with a length of λ0/2) and microstrip antennas. These individual antennas should have a broad beam width, else the scan range of the phased array will be very limited. More details about the radiation characteristics of several antenna types can be found in other chapters. An example of a realized array for radio- astronomy is shown in Fig 6.18, where a 64-element planar array of bow-tie antennas is shown [13]. Now let us assume that the far-field pattern of an isolated array element is given by f~(u, v), which can physically be interpreted as the electric field from generated by an isolated element. The radiation characteristics of an individual array element f~(u, v) are called the element factor or element pattern. Let us for now assume that the radiation characteristics of all array elements are identical. We can then simply apply the superposition principle in order to find the far-field of a planar array of real antenna elements:

K L X X k0[(k−1)dx(u−u0)+(l−1)dy(v−v0)] S~(u, v) = |aj|f~j(u, v)e . (6.44) k=1 l=1

Since we have assumed here that all array elements are identical with f~j = f~, we can rewrite 142 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.18: The one-square metre array (OSMA), a pathfinder for the square kilometer array (SKA) radio telescope [13], [7]. OSMA consists of 64 bow-tie element with a bandwidth of approx. an octave from 1-2 GHz.

(6.44) in the following form:

K L X X k0[(k−1)dx(u−u0)+(l−1)dy(v−v0)] S~(u, v) = f~(u, v) |aj|e . (6.45) k=1 l=1 The normalized radiation pattern can be determined from 6.45 by:

|S~(u, v)|2 F (u, v) = . (6.46) 2 |S~(u0, v0)|

6.4 Mutual coupling

In the previous sections we have assumed that mutual electromagnetic interaction between the individual array elements can be neglected. However, in practise the array elements will be placed on an array grid with a typical element spacing of only λ0/2. Since most antenne elements only operate at resonance, the physical size of the antenna elements will not differ that much from the element spacing. As a result, the individual antenna element will interact with each other, causing so-called mutual coupling. Mutual coupling will affect the element pattern of each array element and will have an influence on the input impedance. Since each antenna element will observe a (slightly) different electromagnetic environment, the element pattern and input impedance of each element will be different. A rigorous electromagnectic analysis of mutual coupling in phased- arrays is outside the scope of this book. Numerical EM tools (e.g. CST, HFSS, Momentum/ADS) can be used to determine the mutual coupling. In addition, in [17] a method is provided to analyse mutual coupling in arrays of microstrip antennas. 6.4. MUTUAL COUPLING 143

In this book we will restrict ourselves to the introduction of the key performance parameters that describe the effect of mutual coupling. We will use the concept of microwave networks [15] in which we will use propagating waves along transmission lines instead of currents and voltages. Let us consider the microwave network of Fig. 6.19. This black-box representation is an equivalent microwave network of a phased-array antenna consisting of K ×L elements (see also Fig. 6.15). A forward wave will propagate with complex amplitude aj along the transmission line connected to the input port of each antenna element j. In addition, a reflected wave with complex amplitude bj will propagate along the same transmission line in the opposite direction. The complex amplitudes of both the forward and reflected waves can be directly measured with a vector network analyser

(VNA). When we consider the array in transmit mode, the reflected signal, represented by bj, is unwanted since it reduces the total signal that is radiated by array element j. As an example to

Microwave Network

p p p p p p a b a b a 1 1 j j KxL bKxL

Figure 6.19: Microwave network representation of a phased-array antenna consisting of K × L antenna elements. further explore the concept, we will consider a very small array with only two antenna elements, so K = 2 en L = 1. The relation between the forward and reflected waves at the input of the network can be written in the following form:

     

 b1   S11 S12   a1          =     , (6.47)             b2 S21 S22 a2 where [S] is the de scattering matrix. Note that S11 represents the reflection coefficient at port 1 when port 2 is terminated with a matched load impedance. In practise the matched load impedance will often by equal to 50Ω zijn). Similarly, S22 is the reflection coefficient at port 2 when port 1 is terminated with a matched load impedance. The coefficient S21 is the coupling coefficient between antenna element 1 en 2. It is a result of the electromagnetic interaction between both antennas. Since array antennas are passive, reciprocity applies. As a result S21 = S12. The 144 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS coupling coefficient can be obtained by using

b2 S21 = |a2=0 . (6.48) a1 In case the array is used in transmit mode, the reflected signal at port 1 will include two compo- nents according to

act S11a1 + S12a2 R1 = , (6.49) a1 where a1 and a2 represent the excitation coefficients which have already been introduced in (6.35). Since a1 and a2 both depend on the scan angle of the main beam of the array (θ0, φ0), the act act resulting reflection coefficient R1 will also depend on the scan angle. Therefore, R1 is referred act to as theactive reflection coefficient of array element 1. Similarly, R2 is the active reflection coefficient of array element 2. From (6.49), we can observe that a large coupling coefficient S12 act act can significantly deteriorate R1 and R2 Note that in practise, the active reflection coefficients not only depend on the scan angle, but also on frequency. For an array consisting of K × L elements, the active reflection coefficient of array element j is give by

1 K×L Ract(u , v ) = X S a (u , v ), (6.50) j 0 0 a ij i 0 0 j i=1 where the excitation coefficients ai with i = 1, 2, ... are given by (6.35). Since part of the incident act 2 power will be reflected, a power loss equal to Lj = 1 − |Rj | will occur. Lj is the scan loss and can be interpreted as a reduction of the antenna gain of array element j. The element factor in 6.45 should then be replaced by:

 2 ~ ~ act fj(u, v) = f(u, v) 1 − Rj (u0, v0) . (6.51)

We can also define the active input impedance of the j-th array element as:

V K×L I Zin,act(u , v ) = j = X Z i , (6.52) j 0 0 I ji I j i=1 j where Vj and Ij are voltage and current at the input port of the j-th element and where Zji is the mutual impedance between array element j and i. The relation between the active input impedance and the active reflection coefficient is now given by:

in,act act Zj (u0, v0) − Z0 Rj (u0, v0) = in,act , (6.53) Zj (u0, v0) + Z0 where Z0 is the port impedance, usually Z0 = 50Ω. The effect of mutual coupling on the performance of phased arrays can be best illustrated with some examples. Let us first consider a large X-band phased-array radar with 4096 open-ended waveguide radiators as presented in [28]. The measured coupling coefficients of a small sub-array of 127 elements near the centre of the array and the associated active reflection coefficient are provided in Fig. 6.20. It can be observed that the active reflection coefficient depends on the 6.5. THE EFFECT OF PHASE AND AMPLITUDE ERRORS 145

Figure 6.20: Measured coupling coefficients of the centre part of 127 array elements and the 0 0 corresponding active reflection coefficient in the H-plane (φ0 = 0 ) and E-plane (φ0 = 90 ). More details can be found in [28] scan angle, but remains quite low up to a scan angle of +/ − 600 in both planes. The measured element patterns in the H-plane (φ = 00) is provided in Fig. 6.21. The element patterns fluctuate 0 around a cos θ pattern, resulting in about 3dB of gain reduction at a scan angle of θ0 = 60 . Another example of the effect of mutual coupling is shown in Fig. 6.22. Due to a periodic effect 0 in this array of 1024 folded-dipoles, a so-called blind scan angle occurs at a scan angle of θ0 = 60 . In this case, all power is reflected and as a result, the directivity will be very low. More details can be found in [29].

6.5 The effect of phase and amplitude errors

In this section we will investigate the effect of phase and amplitude errors on the radiation characteristics of phased arrays. Errors in phase and amplitude at the output of each array element will occur due to non-perfect electronics, mechanical tolerances and environmental changes, like a change in temperature. Let us first consider a linear array of isotropic radiators with uniform amplitude distribution (|ak| = 1. In addition, we will assume that the main beam of the array is directed towards broadside (θ0 = u0 = 0). This configuration is shown in Fig. 6.3. Without any errors the array factor is now given by (see also (6.6))

K X S(u) = ek0(k−1)dxu. (6.54) k=1 Let us now first determine the effect of phase errors. Lateron, we will generalize to include 146 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.21: Measured element patterns of the centre part of 127 array elements in the H-plane 0 0 (φ0 = 0 ) and E-plane (φ0 = 90 ). More details can be found in [28]

Figure 6.22: Photo and the measured element pattern of the centre element of a 1024 phased-array of folded dipoles operating at L-band frequencies. More details can be found in [29] amplitude errors as well. Phase errors will affect the main beam (directivity, beamwidth) and the sidelobes. 6.5. THE EFFECT OF PHASE AND AMPLITUDE ERRORS 147

As a first step we will investigate the main beam. Assume that the applied phase of each element has an error, say δk according to a Gaussian or uniform probability density function with mean value 0 (since we are considering the non-scanning case here first). The array factor at u = 0 can now be written as:

K K K X k0(k−1)dxu−δk X −δk X Smδ = e |u=0 = e = [cos δk −  sin δk] . (6.55) k=1 k=1 k=1

Assume that δk is small, so the sum over the sin δk terms can be neglected, resulting in:

K K  1  1 K S = X cos δ ≈ X 1 − δ2 = K − X δ2, (6.56) mδ k 2 k 2 k k=1 k=1 k=1 where we have applied a Taylor expansion of cos δk. The main beam average power now equals:

!2 1 K |S |2 = K − X δ2 mδ 2 k k=1 (6.57) K 1 K = K2 − K X δ2 + X δ2δ2, k 4 k l k=1 k,l

2 2 Assume that the variance of δk of all elements is equal δk = δ . This implies that for large K and small δ2:

2 2 4 2 δkδl = δ  4Kδ , K 2 2 X 2 (6.58) |Smδ| = K − K δk k=1   = K2 1 − δ2 , since the last term in (6.57) can be neglected. As a second step, we will determine the effect of phase errors on the sidelobes. Let us assume that we are at a null of the error-free radiation pattern. For the non-scanning case the array factor is now written as:

K K X k0(k−1)dxu −δk X βk Ssδ = e e = e [cos δk −  sin δk] , (6.59) k=1 k=1 with βk = k0(k − 1)dxu. Since we are at a null of the error-free pattern, the cos δk term vanishes:

K K X βk X βk Ssδ = = − e sin δk ≈ − e δk. (6.60) k=1 k=1 148 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

We have now transformed the phase-error in an amplitude error occuring in the error-free array pattern. The average power at a null of a sidelobe now becomes:

" K #" K # 2 X β X −β |Ssδ| = e k δk e l δl k=1 l=1 K X X 2 (βk−βl) (6.61) = δk + δkδle k=1 k6=l

= Kδ2,

2 2 with again δk = δ and we have assumed that δk and δl are uncorrelated. The average normalized sidelobe power due to phase errors at a null of the error-free array pattern is now given by:

2 2 2 2 |Ssδ| Kδ δ Rp = =   ≈ . (6.62) 2 2 2 K |Smδ| K 1 − δ

From (6.62) we can conclude that large arrays can tolerate much larger phase errors for a given required sidelobe level. The is due to the fact that we assumed that the errors are uncorrelated. The directivity and associated antenna gain will be reduced in case of phase errors. Starting from the definition of the directivity (6.22), we can determine the directivity in case of errors De of a linear array with uniform illumination with u0 = θ0 = 0 as:

2 P |Smδ| De = = P /4π 2 2 t |Sideal(u)| + |Ssδ(u)| 2   K 1 − δ2 1 − δ2 = = K (6.63) K + Kδ2 1 + δ2 1 ≈ K . 1 + δ2

2 In (6.63), the average radiated power |Sideal(u)| of an error-free linear array with uniform tapering (|ak| = 1) is equal to K. The normalized directivity loss is then:

D 1 e = , (6.64) D 1 + δ2 since the directivity of an ideal linear array with uniform tapering is D = K. Electronic phase shifting is commonly realized using discrete settings. Let us consider a phase shifter with P bits. The total number of settings is 2P . The phase incremental phase step [radians] is:

2π ∆Ψ = . (6.65) k 2P 6.5. THE EFFECT OF PHASE AND AMPLITUDE ERRORS 149

Assuming an uniformly-distributed random phase error in the range [∆Ψk/2, ∆Ψk/2], we find that the variance is given by:  2 2π 2 P π δ2 = 2 = . (6.66) 12 3(22P ) Example: 4-bit phase shifter. Let us consider a 4-bit phase shifter used in a linear array of K = 64 elements with uniform tapering. The phase step is 22.50 and the corresponding maximum phase error is now ±11.250. We now find that the variance is given by:

π2 δ2 = = = 0.0129 [rad2], (6.67) 3(22P ) which corresponds to an average phase error of 6.50 (rms). The directivity loss and the average sidelobe level (SLL) at a null of the error-free pattern can be found by:

2 2 Kδ Rp =   = −36.9 [dB], K2 1 − δ2 (6.68) D 1 e = = −0.0557 [dB]. D 1 + δ2 The corresponding radiation pattern is shown in Fig. 6.23. The ideal pattern has low sidelobes using a Taylor tapering with 30 dB peak sidelobe level.

Figure 6.23: Effect of phase errors using 4-bit phase shifter on the radiation pattern of a linear array with uniform taper with K = 64. The ideal pattern is designed to have a peak sidelobe level (SLL) of 30 dB using a Taylor taper with n = 8. 150 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

As a final step we can generalize to a planar array with both amplitude and phase errors and where the the main beam is scanned towards any direction. Let us consider a planar array with

K ×L elements with phase and amplitude error of element j given by δj and ∆j, respectively. We again will assume that the variance of the phase and amplitude errors of all elements are identical and given by δ2 and ∆2, respectively. It can be shown that the average sidelobe level at a null of the error-free pattern and the directivity loss are given by:

2 2 2 δ + ∆ Rp = , D0 (6.69) D 1 e = , D0 1 + δ2 + ∆2

where D0 is the directivity of the array. Fig. 6.24 provides design guidelines to determine the required amplitude and phase accuracy of the electronic circuits used in a phased array. Three array sizes are compared with K = 64, 1024, 4096 elements.

Figure 6.24: Design guidelines to determine the required accuracy of the phase shifter and variable gain amplifier used in a phased array with K × L antenna elements. Random errors are assumed.

6.6 Noise Figure and Noise Temperature

Let us first do a quick review of noise in microwave receivers by considering the configuration shown in Fig. 6.25. A resistor is connected to the input of an amplifier. The maximum noise power N delivered by a resistor R operating at the environmental temperature T0 is N = kT0B, where k = 1.38 × 10−23 [J/K] is Boltmann’s constant and B [Hz] is the frequency bandwidth of the system. The amplifier is characterized by its gain G and noise figure, which is defined as the 6.6. NOISE FIGURE AND NOISE TEMPERATURE 151

Figure 6.25: Noise in microwave receivers. ratio of the signal to noise ratio at the input versus the signal to noise ratio at the output:   Si Ni F =   So N  o  Si (6.70) kT0B =   GSi No N = o . GkT0B amp Therefore, the additional noise ∆Ni added by the amplifier seen at the input of the amplifier is now:

N − GN ∆N amp = o i = (F − 1)kT B. (6.71) i G 0

The noise power Na from a single antenna element with antenna noise temperature Ta is equal to Na = kTaB. The antenna noise temperature can be determined by integrating the gain function of the antenna over the sky-noise distribution:

1 ZZ T = T (Ω)G (Ω)dΩ, (6.72) a 4π a where Ω is the solid angle with dΩ = sin θdθdφ. Note that Ta is often much lower than the environmental temperature due to the weighting of the sky-noise temperature distribution T (Ω) with the gain G(Ω) of the antenna. For example in satellite TV at Ku-band (11-14 GHz), a typical value is Ta = 100 K. Now consider a phased array with K elements with uniform amplitude taper, as illustrated in Fig 6.26. Each antenna element is connected to an electronic amplifier with gain G and noise figure

F . The noise power received by a single antenna element k is Nka = kTkaB. Let us assume that Tka = Ta. The K : 1 power combiner coherently combines the antenna noise power contributions 152 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.26: Array with K elements. Each antenna element is connected to a low-noise amplifier with gain G and noise figure F . from all elements, since (worst-case) the wanted signal and the antenna noise could be co-located at the same spherical coordinate (u0, v0). The total output noise power due to the antenna noise is than:

T (6.73) Na = KGNka = kTaBKG.

The noise from the electronics at the input of the power combiner is given by:

(6.74) Ne = kT0B(F − 1)G.

The transmitted noise due to the electronics at the output of the power combiner now becomes:

kT B(F − 1)G N o = 0 . (6.75) e K And as a result, the total noise at the output of the combiner will become:

T o 0 (6.76) Ne = KNe = kT0B(F − 1)G.

The signal power at the output of the combiner is:

(6.77) So = KGSa, 6.7. SPARSE ARRAYS 153 where Sa is the signal power received by a single array element. The signal-to-noise ratio at the output of the combiner is therefore given by:

  S So KGSa = T T o = N out Na + Ne kTaBKG + kT0B(F − 1)G (6.78) S = a . kTaB + kT0B(F − 1)/K From (6.78) we can conclude that the noise performance of a phased array is identical to the case of a single equivalent large antenna (e.g. a reflector antenna) with antenna gain equal to the antenna gain of the phased array. The equivalent antenna is connected to a low-noise amplifier with a gain G and noise figure F as illustrated in Fig. 6.27.

Figure 6.27: Noise in a phased-array can be modelled by considering an equivalent single antenna with the same antenna gain and connected to a single low-noise amplifier with gain G and noise figure F . In case of an ideal array with uniform taper the antenna gain is equal to K.

6.7 Sparse arrays

Most application of phased-arrays do not allow grating lobes. In radar applications, grating lobes would create confusion about the position of the target and it would spread the radiated power in transmit mode into multiple directions. In telecommunication applications, grating lobes could allow strong interfering signals to saturate the receiver electronics. However, some applications could allow grating lobes to appear. An example is radio astronomy. Sparse arrays are arrays of which the inter-element distance (dx, dy) is much larger than 0.5λ0. Fig. 6.28 illustrates the basic concept of sparse arrays. Sparse array can have a regular grid or irregular grid. Fig. 6.29 shows the radiation pattern of a linear array with K = 32 elements placed on a dense versus sparse regular grid. The sparse array clearly has a much narrower main beam. The price to pay are the appearance of several grating lobes. A way to remove the unwanted grating lobes in sparse arrays is to use an irregular array grid. In this case the radiated power in the grating lobes are spread out over the entire visible space. As a result, the average sidelobe level will be higher and will 154 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.28: Dense regular versus sparse irregular arrays

Figure 6.29: Radiation pattern of a dense regular array and sparse regular array. A linear array of 32 elements with uniform tapering is assumed. depend on the amount of array elements. An example of a very efficient irregular sparse array configuration is the sun-flower topology, as shown in Fig. 6.30. The average element distance in both the regular and sun-flower configuration in Fig. 6.30 is about 2λ0. Fig. 6.31 shows the corresponding radiation patterns of both arrays. Clearly, the sunflower array suppresses the grating lobes at the expense of a higher average sidelobe level. When the element distribution in a sparse array is random or quasi-random, it can be shown that the average sidelobe level for large arrays with K × L elements, is approximated by:

1 SLL = . (6.79) KL

So in case of a sparse-random array with 1000 elements, the average SLL will be -30 dB below the main beam gain. 6.8. CALIBRATION OF PHASED-ARRAYS 155

Figure 6.30: Sparse regular array (left) and sparse irregular array (sun-flower arrangement). Both arrays have 289 elements and have an average element spacing of about 2λ0.

Figure 6.31: Radiation pattern in the (u, v)-plane of the sparse regular and sun-flower array. Main 0 0 beam is scanned towards (θ0, φ0) = (30 , 0 ). An uniform amplitude tapering is used.

6.8 Calibration of phased-arrays

Amplitude and phase errors will occur in phased-array systems. These errors can have a systematic time-invariant nature, but can also be caused by changes in the environmental conditions, e.g. temperature changes. Calibration must be used to correct for these errors. A schematic diagram of the calibration principle often used in modern phased arrays [7] is shown in Fig. 6.32. Two types of signals can be injected into each array element: an external sigal generated by a far-field FF cal source Hk , or a local calibration signal Hk injected via a special calibration network within the array or by means of injection via mutual coupling with a special calibration element, as illustrated in Fig. 6.32. The first step in the calibration procedure is to relate the received signals 156 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS

Figure 6.32: Calibration principle used in phased arrays. Only element k is shown. at the output of each array element from a test signal of one of the calibration elements to the received signals due to one or more incident plane waves from a far-field source. This part of the calibration procedure is called off-line calibration, since it needs to be done only once in a high-quality anechoic facility, e.g. by using a near-field scanner or compact antenna range. The element calibration coefficients are calculated and stored in a look-up table. With ak the received signal from the far-field source and bk the received signal from the calibration element, we obtain the following calibration coefficients in our look-up table:

FF com FF ak Hk Hk Hk (6.80) ck = = cal com = cal . bk Hk Hk Hk

The second step is to measure the amplitude ad phase at the output of each array element due to a signal from one of the calibration elements. This part is called on-line calibration, since it can be done while the system is operation in the application environment. The measured signal from the calibration element is now:

˜ cal ˜ com (6.81) bk = Hk Hk .

By combining the results from both calibration steps, we can re-construct the signal transfer from a far-field source to array element k:

FF ! FF ˜ com cal ˜ com Hk ˜ (6.82) a˜k = Hk Hk = Hk Hk cal = bk ck. Hk

Phased arrays often consist of many elements, e.g. 4096. Therefore, the on-line calibration could take a lot of time. Several techniques have been proposed in literature to overcome this problem. An example is the multi-element phase togging (MEP) technique [7], in which the FFT principle is used to calibrate groups of array elements simultaneously. 6.9. FOCAL-PLANE ARRAYS 157

6.9 Focal-plane arrays

Phased-arrays often require a lot of individual antenna elements and associated electronic circuits, leading to high cost and high power consumption. Several applications require a high antenna gain with only a limited scan capabilities. In this case, the hybrid concept of focal-plane arrays (FPAs) is very interesting. FPAs combine the benefits of phased-arrays and traditional reflector-based solutions by offering a high antenna gain, relative low costs and electronic beam scanning over a limited field-of-view (FoV). Therefore, it has become an interesting alternative for conventional horn-fed reflector antennas in a number of applications, e.g. in radio astronomy and in satellite communication. Moreover, emerging applications such as point-to-point wireless communications, 5G new-radio millimeter-wave (mm-wave) wireless and low-cost Ka-band multi-function radars could be areas in which FPAs can play a major role. Fig. 6.33 shows the concept of a basic FPA system. A phased-array feed (PAF) is placed in the focal-plane in front of a parabolic reflector antenna with diameter D and focal length F . When an incident plane wave illuminates the reflector, the location of the beam focus moves on the phased-array feed depends on the scan- angle and moves along the array feed. Therefore, for each scan angle a limited number of array elements are active. The antenna gain of an FPA is mainly determined by the size of the reflector and not by the number of array elements in the PAF. In this way, a high antenna gain can be realized with electronic beam-scanning capabilities. For large F/D ratio and small scan angles

Figure 6.33: Focal-plane array (FPA) concept in which a phased-array feed is placed in the focal- plane in front of a parabolic reflector antenna, from [30]. When an incident plane wave illuminates the reflector, the location of the beam focus moves along the phased-array feed with varying angle of incidence (scan angle). the electric field distribution in the focal plane of a prime-focus parabolic reflector antenna can 158 CHAPTER 6. PHASED ARRAYS AND SMART ANTENNAS be approximated by:

2J (k r sin Ψ ) E(r) = 1 0 0 , (6.83) k0r sin Ψ0 where J1 is the Bessel function of the first kind with order 1, Ψ0 = π/4, the subtended angle of the reflector, k0 is the free-space wavenumber and r is the distance from the centre of the focal plane (aperture radius), see Fig. 6.33. The field distribution in the focal plane is illustrated in the example of Fig. 6.34 in case of broadside incidence of a plane wave. The rings in the field distribution are also known as Airy rings. We can clearly see that only a limited number of array elements will be active simultaneously for a particular angle of incidence (scan angle). In case of scanning, the pattern will shift along the focal plane. The aperture efficiency η as a function of

Figure 6.34: (a) Focal-plane array illuminated by a plane wave (broadside incidence), (b) Field distribution in the focal plane according to 6.83 for a classical prime focus parabolic reflector with F/D = 0.6 and f = 30 GHz. Dimensions in focal plane in [m]. Figure from [30] the radius of the PAF feed rp of the FPA can be determined by integrating the electric field in the focal plane and normalize it to the total power Ptot obtained by the reflector:

2π rp Z Z 1 2 η(rp) = |E(r)| drdφ, (6.84) Ptot 0 0 More details on the scanning capabilities and limitations can be found in [30]. Bibliography

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